(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips 5.516 Copyright 1986, 1993 Radical Eye Software %%Title: paper.dvi %%CreationDate: Fri Mar 15 20:39:12 1996 %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o paper.ps paper %DVIPSSource: TeX output 1996.03.15:2038 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{ pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D }B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (/tmp_mnt/home/u2/jordan/papers/xu-neuralComputation94/paper.dvi) @start /Fa 52 122 df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b 12 120 df<000070000000007000000000F800000000F800000000 F800000001FC00000001FC00000003FE00000003FE00000003FE00000006FF000000067F 0000000E7F8000000C3F8000000C3F800000183FC00000181FC00000381FE00000300FE0 0000300FE00000600FF000006007F00000E007F80000FFFFF80000FFFFF800018001FC00 018001FC00038001FE00030000FE00030000FE000600007F000600007F00FFE00FFFF8FF E00FFFF825227EA12A>65 D<00FF8007FFE00F83F01F03F03E03F07E03F07C01E07C0000 FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00301F00600FC0E0 07FF8000FE0014167E9519>99 D<0001FE000001FE0000003E0000003E0000003E000000 3E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0001FC3E0007FF BE000F81FE001F007E003E003E007E003E007C003E00FC003E00FC003E00FC003E00FC00 3E00FC003E00FC003E00FC003E00FC003E007C003E007C003E003E007E001E00FE000F83 BE0007FF3FC001FC3FC01A237EA21F>I<00FE0007FF800F87C01E01E03E01F07C00F07C 00F8FC00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E00003E00181F 00300FC07003FFC000FF0015167E951A>I<03FC1E0FFF7F1F0F8F3E07CF3C03C07C03E0 7C03E07C03E07C03E07C03E03C03C03E07C01F0F801FFF0013FC00300000300000380000 3FFF801FFFF00FFFF81FFFFC3800FC70003EF0001EF0001EF0001EF0001E78003C7C007C 3F01F80FFFE001FF0018217E951C>103 D107 DIII<00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C007CFC00 7EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C3E00F81F01F00F83E007FF C000FE0017167E951C>I<0180000180000180000180000380000380000780000780000F 80003F8000FFFF00FFFF000F80000F80000F80000F80000F80000F80000F80000F80000F 80000F80000F80000F81800F81800F81800F81800F81800F830007C30003FE0000F80011 207F9F16>116 D119 D E /Fc 1 51 df<7FFFFF80FFFFFF80C000 0180C0000180C0000180C0000180C0000180C0000180C0000180C0000180C0000180C000 0180C0000180C0000180C0000180C0000180C0000180C0000180C0000180C0000180C000 0180C0000180C0000180FFFFFF807FFFFF8019197C9B22>50 D E /Fd 23 122 df<0018007000E001C00380038007000E000E001E001C003C003C00780078 0078007800F800F000F000F000F000F000F000F000F000F000F80078007800780078003C 003C001C001E000E000E0007000380038001C000E0007000180D2D7DA114>40 DI<387CFEFEFE7C3807077C 860F>46 D<00E00001E0000FE000FFE000F3E00003E00003E00003E00003E00003E00003 E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003 E00003E00003E00003E00003E000FFFF80FFFF80111D7C9C1A>49 D<07F0001FFE00383F007C1F80FE0FC0FE0FC0FE0FE0FE07E07C07E03807E0000FE0000F C0000FC0001F80001F00003E0000780000F00000E00001C0000380600700600E00601C00 E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131D7D9C1A>I<01FC0007FF000E0F801E0FC03F 07E03F07E03F07E03F07E01E0FC0000FC0000F80001F0001FC0001FC00000F800007C000 03E00003F00003F83803F87C03F8FE03F8FE03F8FE03F0FC03F07807E03C0FC01FFF8003 FC00151D7E9C1A>I<0007FC02003FFF0E00FE03DE03F000FE07E0003E0FC0001E1F8000 1E3F00000E3F00000E7F0000067E0000067E000006FE000000FE000000FE000000FE0000 00FE000000FE000000FE0000007E0000007E0000067F0000063F0000063F00000C1F8000 0C0FC0001807E0003803F0007000FE01C0003FFF800007FC001F1F7D9E26>67 D76 D80 D82 D<7FFFFFFC7FFFFFFC7C07E07C7007E01C6007E00C6007E00CE007E00EC007E006C007E0 06C007E006C007E0060007E0000007E0000007E0000007E0000007E0000007E0000007E0 000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0 000007E00003FFFFC003FFFFC01F1E7E9D24>84 D<07FC001FFF003F0F803F07C03F03E0 3F03E00C03E00003E0007FE007FBE01F03E03C03E07C03E0F803E0F803E0F803E0FC05E0 7E0DE03FF8FE0FE07E17147F9319>97 D<01FE0007FF801F0FC03E0FC03E0FC07C0FC07C 0300FC0000FC0000FC0000FC0000FC0000FC00007C00007E00003E00603F00C01F81C007 FF0001FC0013147E9317>99 D<01FE0007FF800F83C01E01E03E00F07C00F07C00F8FC00 F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00003E00181E00180F807007FFE000FF 8015147F9318>101 D<001F8000FFC001F3E003E7E003C7E007C7E007C3C007C00007C0 0007C00007C00007C000FFFC00FFFC0007C00007C00007C00007C00007C00007C00007C0 0007C00007C00007C00007C00007C00007C00007C00007C00007C0003FFC003FFC001320 7F9F10>I104 D108 DII<01FF 0007FFC01F83F03E00F83E00F87C007C7C007CFC007EFC007EFC007EFC007EFC007EFC00 7E7C007C7C007C3E00F83E00F81F83F007FFC001FF0017147F931A>I114 D<0FE63FFE701E600EE006 E006F800FFC07FF83FFC1FFE03FE001FC007C007E007F006F81EFFFCC7F010147E9315> I121 D E /Fe 14 116 df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f 8 117 df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g 4 108 df0 D<00000600001E0000780001E00007 80001E0000780001E0000780001E0000780000E000007800001E000007800001E0000078 00001E000007800001E000007800001E0000060000000000000000000000000000000000 007FFFFCFFFFFE171F7D971E>20 DI<8040C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0C0C0C0C0C0C080400A267C9B13>107 D E /Fh 5 115 df<0004000C0018001800 1800300030003000600060006000C000C000C00180018001800300030003000600060006 000C000C000C00180018001800300030003000600060006000C000C0000E257E9B13>61 D<1F8003800380038007000700070007000E000E000E0E0E331C471C831D001E003C003F 8039C038E070E270E270E270E4E0646038101A7E9914>107 D<03E006101C18381C300C 701C601CE01CE01CE018E038C030E07060E021801F000E107E8F13>111 D<03840C5C1838383830387038E070E070E070E070C0E0C0E0C0E0E1E063C03DC001C001 C003800380038003803FE00E177E8F11>113 D<38704D988F388E188E008E001C001C00 1C001C003800380038003800700030000D107E8F11>I E /Fi 42 122 df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j 8 62 df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k 2 4 df0 D<0C000C008C40EDC07F800C007F80EDC08C400C000C000A0B7D8B10>3 D E /Fl 3 66 df0 D<020002000200C218F2783AE00F80 0F803AE0F278C2180200020002000D0E7E8E12>3 D<0000180000003800000038000000 7800000078000000F8000000B8000001B80000013800000338000006380000063800000C 380000183C0000183C0000301C00006FFC00007FFC0000C01C0001801C0083801E00C700 1E00FE000F807C000F003800000019197F971C>65 D E /Fm 16 113 df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n 19 121 df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o 44 121 df<007C0001C3000701810E01C1 1E00C11C00E23C00E27800E27800E47800E4F000E8F000F0F000F0F000E0F000E07000E0 7003E030046118383207C01C18147E931D>11 D<03E0040FF0081FF8083FF810301C1040 0C10C0042080042000024000024000028000028000028000030000030000030000020000 0200000200000600000600000600000C00000C00000C00000C0000180000180000180000 1000161E7F9318>13 D<001E0000610000C0800180800180000380000300000380000380 0003C00001F00000F800007C0001FC00070E000E0E001E06001C06003C06007806007806 00780600F00400F00400F00C00F00800F008007018007010003020001840000F80001120 7E9F14>I<1E07C023186023A03043C0304380384380388700700700700700700700700E 00E00E00E00E00E00E00E01C01C01C01C01C01C01C01C038038018038000038000038000 0700000700000700000700000E00000E00000E00000C00151E7E9317>17 D<001E0000630000C38001C1800381800301C00701C00F01C00E01C01E03C01C03C03C03 C03C03C03C03C07807807FFF807FFF80780780F00F00F00F00F00F00F00E00F01E00E01C 00E03C00E03800E0300060700060E00070C0003180001E000012207E9F15>I<06007007 01F80E02F00E0C600E10000E20001C40001C80001F00001FF800381E0038070038070038 0384700708700708700708700310E003106001E016147E931A>20 D<07000001C00000E00000E00000F000007000007000007800003800003800003C00001C 00001C00001E00000E00000E00000F00000700000F0000378000638000C38001C3C00381 C00701C00E01E01C00E03800E07000F0F00070E00070C0003815207D9F1B>I<0FFFFC1F FFFC3FFFFC608200C08400808400018400010400010C00030C00030C00020C00060C0006 0C000E0C000C0E001C0E001C0E00380F0018060016147E931A>25 D<007FFE01FFFE07FFFE0F07801E03801C01C03801C07001C07001C07001C0E00380E003 80E00380E00700E00700E00E00600C003018001860000F800017147E931A>27 D<70F8F8F87005057C840D>58 D<70F8FCFC74040404080810102040060E7C840D>I<00 0001C00000078000001E00000078000001E00000078000000E00000038000000F0000003 C000000F0000003C000000F0000000F00000003C0000000F00000003C0000000F0000000 380000000E0000000780000001E0000000780000001E0000000780000001C01A1A7C9723 >I<000100030003000600060006000C000C000C00180018001800300030003000600060 006000C000C000C00180018001800300030003000600060006000C000C000C0018001800 1800300030003000600060006000C000C000C000102D7DA117>II<001F000060C0 00806001003002001803801803C01C07801C03001C00001E00001E00001E003E1E00C11E 03809C07007C0E007C1C007C3C007C3800787800787800787800F0F000F0F000E0F000E0 F001C0F001C0F00380700300700600380C001C380007E00017227EA018>64 D<000002000000060000000E0000000E0000001E0000001F0000002F0000002F0000004F 0000008F0000008F0000010F0000010F0000020F0000040F0000040F0000080F80000807 80001007800020078000200780007FFF8000400780008007800180078001000780020007 80020007C0040003C00C0003C01E0007C0FF807FFC1E207E9F22>I<00FFFFE0000F0078 000F003C000F001C000F001E001E001E001E001E001E001E001E001E003C003C003C003C 003C0078003C00F0007803C0007FFF80007803C0007801E000F000F000F000F000F000F0 00F0007001E000F001E000F001E000F001E000E003C001E003C003C003C0038003C00F00 07801E00FFFFF0001F1F7E9E22>I<0000FE0200078186001C004C0038003C0060003C00 C0001C01C0001803800018070000180F0000181E0000101E0000103C0000003C00000078 000000780000007800000078000000F0000000F0000000F0000000F0000000F000008070 00008070000080700001003800010038000200180004000C001800060020000381C00000 FE00001F217E9F21>I<00FFFFFF000F000E000F0006000F0002000F0002001E0002001E 0002001E0002001E0002003C0404003C0400003C0400003C0C0000781800007FF8000078 18000078180000F0100000F0100000F0100000F0000401E0000801E0000801E0001001E0 001003C0002003C0006003C0004003C001C0078007C0FFFFFF80201F7E9E22>69 D<00FFF9FFF0000F801F00000F001E00000F001E00000F001E00001E003C00001E003C00 001E003C00001E003C00003C007800003C007800003C007800003C007800007800F00000 7FFFF000007800F000007800F00000F001E00000F001E00000F001E00000F001E00001E0 03C00001E003C00001E003C00001E003C00003C007800003C007800003C007800003C007 800007C00F8000FFF8FFF800241F7E9E26>72 D<00FFFC000F80000F00000F00000F0000 1E00001E00001E00001E00003C00003C00003C00003C0000780000780000780000780000 F00000F00000F00000F00001E00001E00001E00001E00003C00003C00003C00003C00007 C000FFFC00161F7F9E14>I<001FFF0000F80000F00000F00000F00001E00001E00001E0 0001E00003C00003C00003C00003C0000780000780000780000780000F00000F00000F00 000F00001E00001E00301E00781E00F83C00F83C00F0780080700040E00021C0001F0000 18207D9E19>I<00FFF80FF8000F8003E0000F000380000F000200000F000400001E0008 00001E002000001E004000001E008000003C010000003C040000003C080000003C180000 007838000000787C000000793C0000007A3C000000F41E000000F81E000000F01E000000 F00F000001E00F000001E00F000001E007800001E007800003C007800003C003C00003C0 03C00003C003C00007C003E000FFFC3FFC00251F7E9E27>I<00FF00001FF0000F00003F 00000B80003E00000B80005E00000B80005E0000138000BC00001380013C00001380013C 00001380023C000023800278000023800478000023800878000021C00878000041C010F0 000041C020F0000041C020F0000041C040F0000081C041E0000081C081E0000081C101E0 000081C101E0000100E203C0000100E203C0000100E403C0000100E803C0000200E80780 000200F00780000200F00780000600E00780000F00C00F8000FFE0C1FFF8002C1F7E9E2C >77 D<00FF803FF0000F800780000F800200000BC00200000BC002000013C004000011E0 04000011E004000011E004000020F008000020F008000020F80800002078080000407810 0000403C100000403C100000403C100000801E200000801E200000801E200000800F2000 01000F400001000F4000010007C000010007C00002000780000200038000020003800006 000380000F00010000FFE0010000241F7E9E25>I<00FFFFC0000F0070000F0038000F00 1C000F001E001E001E001E001E001E001E001E001E003C003C003C003C003C0078003C00 70007800E000780380007FFE000078000000F0000000F0000000F0000000F0000001E000 0001E0000001E0000001E0000003C0000003C0000003C0000003C0000007C00000FFFC00 001F1F7E9E1D>80 D<0FFFFFFC1E03C0381803C0181003C0082003C00820078008600780 084007800840078008800F0010000F0000000F0000000F0000001E0000001E0000001E00 00001E0000003C0000003C0000003C0000003C0000007800000078000000780000007800 0000F0000000F0000000F0000000F0000001F000007FFFC0001E1F7F9E1B>84 D<7FFC1FF807C003C00780010007800100078001000F0002000F0002000F0002000F0002 001E0004001E0004001E0004001E0004003C0008003C0008003C0008003C000800780010 00780010007800100078001000F0002000F0002000F0002000F0004000F0004000700080 007001000030020000380400000C18000007E000001D207C9E1F>II<007C01C207010E0F1E0F1C0E3C04780078 007800F000F000F000F000F00070017002300418380FC010147E9314>99 D<007C01C207010E011C013C013802780C7BF07C00F000F000F000F00070007001700230 04183807C010147E9315>101 D<00007C0000CE00019E00039E00030C00070000070000 0700000700000E00000E00000E0000FFF0000E00000E00001C00001C00001C00001C0000 1C0000380000380000380000380000380000700000700000700000700000700000E00000 E00000E00000E00000C00001C000318000798000F300006200003C000017297E9F16>I< 01E0000FE00001C00001C00001C00001C000038000038000038000038000070000070000 071F000761800E80C00F00C00E00E00E00E01C01C01C01C01C01C01C01C0380380380380 380380380704700708700E08700E10700610E006206003C016207E9F1A>104 D<00E001E001E000C000000000000000000000000000000E001300238043804380438087 00070007000E000E001C001C001C20384038403840388019000E000B1F7E9E10>I<0000 C00001E00001E00001C0000000000000000000000000000000000000000000001E000063 00004380008380010380010380020700000700000700000700000E00000E00000E00000E 00001C00001C00001C00001C0000380000380000380000380000700000700030700078E0 00F1C0006380003E00001328819E13>I<01E0000FE00001C00001C00001C00001C00003 80000380000380000380000700000700000701E00706100E08700E10F00E20F00E40601C 80001D00001E00001FC000387000383800383800381C20703840703840703840701880E0 1880600F0014207E9F18>I<03C01FC0038003800380038007000700070007000E000E00 0E000E001C001C001C001C0038003800380038007000700070007100E200E200E200E200 640038000A207E9F0E>I<1E07C07C00231861860023A032030043C03403004380380380 438038038087007007000700700700070070070007007007000E00E00E000E00E00E000E 00E00E000E00E01C101C01C01C201C01C038201C01C038401C01C0184038038018801801 800F0024147E9328>I<03C1E004621804741C08781C08701E08701E10E01E00E01E00E0 1E00E01E01C03C01C03C01C03C01C0380380780380700380E003C1C0072380071E000700 000700000E00000E00000E00000E00001C00001C0000FFC000171D819317>112 D<1E1E0023210023C38043C7804387804383008700000700000700000700000E00000E00 000E00000E00001C00001C00001C00001C000038000018000011147E9315>114 D<007C018203010603060706060E00078007F803FC01FE001F00077007F006F006E00440 0820301FC010147E9315>I<00C000E001C001C001C001C003800380FFF8038007000700 070007000E000E000E000E001C001C001C001C10382038203820384018800F000D1C7F9B 10>I<0F00601180702180E021C0E041C0E04380E08381C00701C00701C00701C00E0380 0E03800E03800E03840E07080C07080C07080E0F1006131003E1E016147E931A>I<03C1 C00C62201034701038F02038F020386040700000700000700000700000E00000E00000E0 0000E02061C040F1C040F1C080E2C080446300383C0014147E931A>120 D E /Fp 11 62 df<00FC000303000E01C01C00E0380070300030700038700038E0001C E2011CE3FF1CE3FF1CE3FF1CE2011CE0001C6000187000383000303800701C00E00E01C0 03030000FC0016177E961B>2 D6 D<0102040C1818303070606060E0E0E0E0E0 E0E0E0E0E060606070303018180C04020108227D980E>40 D<8040203018180C0C0E0606 06070707070707070707070606060E0C0C18183020408008227E980E>I<003000003000 003000003000003000003000003000003000003000003000003000FFFFFCFFFFFC003000 00300000300000300000300000300000300000300000300000300000300016187E931B> 43 D<07C018303018701C600C600CE00EE00EE00EE00EE00EE00EE00EE00EE00E600C60 0C701C30181C7007C00F157F9412>48 D<03000700FF0007000700070007000700070007 0007000700070007000700070007000700070007007FF00C157E9412>I<0F8030E04070 8030C038E0384038003800700070006000C00180030006000C08080810183FF07FF0FFF0 0D157E9412>I<0FE030306018701C701C001C00180038006007E000300018000C000E00 0EE00EE00EC00C401830300FE00F157F9412>I<00300030007000F001F0017002700470 08701870107020704070C070FFFE0070007000700070007003FE0F157F9412>I61 D E /Fq 34 123 df<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00 00FE0000FE0000FE007FFFFE7FFFFE7FFFFE17277BA622>49 D<00FF800003FFF0000FFF FC001F03FE003800FF007C007F80FE003FC0FF003FC0FF003FE0FF001FE0FF001FE07E00 1FE03C003FE000003FE000003FC000003FC000007F8000007F000000FE000000FC000001 F8000003F0000003E00000078000000F0000001E0000003C00E0007000E000E000E001C0 01C0038001C0070001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF80FFFFFF80FFFF FF801B277DA622>I<007F800003FFF00007FFFC000F81FE001F00FF003F80FF003F807F 803F807F803F807F801F807F800F007F800000FF000000FF000000FE000001FC000001F8 000007F00000FFC00000FFF0000001FC0000007E0000007F0000007F8000003FC000003F C000003FE000003FE03C003FE07E003FE0FF003FE0FF003FE0FF003FC0FF007FC07E007F 807C007F003F01FE001FFFFC0007FFF00000FF80001B277DA622>I<00000E0000001E00 00003E0000007E000000FE000000FE000001FE000003FE0000077E00000E7E00000E7E00 001C7E0000387E0000707E0000E07E0000E07E0001C07E0003807E0007007E000E007E00 0E007E001C007E0038007E0070007E00E0007E00FFFFFFF8FFFFFFF8FFFFFFF80000FE00 0000FE000000FE000000FE000000FE000000FE000000FE000000FE00007FFFF8007FFFF8 007FFFF81D277EA622>I<0C0003000F803F000FFFFE000FFFFC000FFFF8000FFFF0000F FFE0000FFFC0000FFE00000E0000000E0000000E0000000E0000000E0000000E0000000E 7FC0000FFFF8000F80FC000E003E000C003F0000001F8000001FC000001FC000001FE000 001FE018001FE07C001FE0FE001FE0FE001FE0FE001FE0FE001FC0FC001FC078003F8078 003F803C007F001F01FE000FFFF80003FFF00000FF80001B277DA622>I<0007F000003F FC0000FFFE0001FC0F0003F01F8007E03F800FC03F801FC03F801F803F803F801F003F80 00007F0000007F0000007F000000FF000000FF0FC000FF3FF800FF707C00FFC03E00FFC0 3F00FF801F80FF801FC0FF001FC0FF001FE0FF001FE0FF001FE07F001FE07F001FE07F00 1FE07F001FE03F001FE03F001FC01F801FC01F803F800FC03F0007E07E0003FFFC0000FF F000003FC0001B277DA622>I<00003FF001800003FFFE0380000FFFFF8780003FF007DF 8000FF8001FF8001FE00007F8003FC00003F8007F000001F800FF000000F801FE0000007 801FE0000007803FC0000007803FC0000003807FC0000003807F80000003807F80000000 00FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF80000000 00FF8000000000FF8000000000FF80000000007F80000000007F80000000007FC0000003 803FC0000003803FC0000003801FE0000003801FE0000007000FF00000070007F000000E 0003FC00001E0001FE00003C0000FF8000F800003FF007E000000FFFFFC0000003FFFF00 0000003FF8000029297CA832>67 D69 D<00007FE003000003FFFC0700001FFFFF0F00003FF00FFF00 00FF8001FF0001FE0000FF0003F800003F0007F000003F000FF000001F001FE000000F00 1FE000000F003FC000000F003FC0000007007FC0000007007F80000007007F8000000000 FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000FF8000000000 FF8000000000FF8000000000FF8001FFFFF87F8001FFFFF87F8001FFFFF87FC00000FF00 3FC00000FF003FC00000FF001FE00000FF001FE00000FF000FF00000FF0007F00000FF00 03F80000FF0001FE0000FF0000FF8001FF00003FF007BF00001FFFFF1F000003FFFE0F00 00007FF003002D297CA836>71 D73 D77 D<7FFFFFFFFFC07FFFFFFFFFC07FFFFFFFFFC07F803FC03F C07E003FC007C078003FC003C078003FC003C070003FC001C0F0003FC001E0F0003FC001 E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC000 0000003FC0000000003FC0000000003FC00000007FFFFFE000007FFFFFE000007FFFFFE0 002B287EA730>84 D<01FF800007FFF0000F81F8001FC07E001FC07E001FC03F000F803F 8007003F8000003F8000003F8000003F80000FFF8000FFFF8007FC3F800FE03F803F803F 803F003F807F003F80FE003F80FE003F80FE003F80FE003F807E007F807F00DF803F839F FC0FFF0FFC01FC03FC1E1B7E9A21>97 DI<001FF80000FFFE0003F01F0007E03F800FC0 3F801F803F803F801F007F800E007F0000007F000000FF000000FF000000FF000000FF00 0000FF000000FF000000FF0000007F0000007F0000007F8000003F8001C01F8001C00FC0 038007E0070003F01E0000FFFC00001FE0001A1B7E9A1F>I<00003FF80000003FF80000 003FF800000003F800000003F800000003F800000003F800000003F800000003F8000000 03F800000003F800000003F800000003F800000003F800000003F800001FE3F80000FFFB F80003F03FF80007E00FF8000FC007F8001F8003F8003F8003F8007F0003F8007F0003F8 007F0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800 FF0003F8007F0003F8007F0003F8007F0003F8003F8003F8001F8003F8000F8007F80007 C00FF80003F03BFF8000FFF3FF80003FC3FF80212A7EA926>I<003FE00001FFF80003F0 7E0007C01F000F801F801F800F803F800FC07F000FC07F0007C07F0007E0FF0007E0FF00 07E0FFFFFFE0FFFFFFE0FF000000FF000000FF0000007F0000007F0000007F0000003F80 00E01F8000E00FC001C007E0038003F81F0000FFFE00001FF0001B1B7E9A20>I<0007F0 003FFC00FE3E01F87F03F87F03F07F07F07F07F03E07F00007F00007F00007F00007F000 07F00007F000FFFFC0FFFFC0FFFFC007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F0007FFF807FFF807FFF80182A7EA915>I<00FF81F003FFE7F80FC1FE7C1F80 FC7C1F007C383F007E107F007F007F007F007F007F007F007F007F007F007F007F003F00 7E001F007C001F80FC000FC1F8001FFFE00018FF800038000000380000003C0000003E00 00003FFFF8001FFFFF001FFFFF800FFFFFC007FFFFE01FFFFFF03E0007F07C0001F8F800 00F8F80000F8F80000F8F80000F87C0001F03C0001E01F0007C00FC01F8003FFFE00007F F0001E287E9A22>II<07000FC01FE03FE03FE03FE01FE00FC007000000000000000000 000000000000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7DAA14>I108 DII<003FE00001FF FC0003F07E000FC01F801F800FC03F800FE03F0007E07F0007F07F0007F07F0007F0FF00 07F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F87F0007F07F00 07F03F800FE03F800FE01F800FC00FC01F8007F07F0001FFFC00003FE0001D1B7E9A22> II114 D<03FE300FFFF01E03F03800F0700070F00070F00070F80070FC0000FFE0007F FE007FFF803FFFE01FFFF007FFF800FFF80003FC0000FC60007CE0003CF0003CF00038F8 0038FC0070FF01E0F7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000F000 00F00000F00001F00003F00003F00007F0001FFFF0FFFFF0FFFFF007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F03807F038 07F03807F03807F03807F03803F03803F87001F86000FFC0001F8015267FA51B>IIIII<3FFFFF803FFFFF803F007F003C00FE003801FE 007803FC007803F8007007F800700FF000700FE000001FC000003FC000007F8000007F00 0000FF000001FE038001FC038003F8038007F803800FF007800FE007801FE007003FC00F 003F801F007F007F00FFFFFF00FFFFFF00191B7E9A1F>122 D E /Fr 1 81 df<01FFFF00003C03C0003800E0003800F00038007000380070007000F00070 00F0007000F0007000E000E001E000E003C000E0078000E01E0001FFF00001C0000001C0 000001C00000038000000380000003800000038000000700000007000000070000000700 00000F000000FFE000001C1C7E9B1B>80 D E /Fs 73 123 df<007E1F0001C1B1800303 E3C00703C3C00E03C1800E01C0000E01C0000E01C0000E01C0000E01C0000E01C000FFFF FC000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01 C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0007F87FC001A1D 809C18>11 D<007E0001C1800301800703C00E03C00E01800E00000E00000E00000E0000 0E0000FFFFC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C00E01C00E01C07F87F8151D809C17>I<003F07E00001C09C 18000380F018000701F03C000E01E03C000E00E018000E00E000000E00E000000E00E000 000E00E000000E00E00000FFFFFFFC000E00E01C000E00E01C000E00E01C000E00E01C00 0E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E 00E01C000E00E01C000E00E01C000E00E01C000E00E01C007FC7FCFF80211D809C23>14 D<6060F0F0F8F86868080808080808101010102020404080800D0C7F9C15>34 D<00E0000001900000030800000308000007080000070800000708000007080000071000 0007100000072000000740000003C03FE003800F00038006000380040005C0040009C008 0010E0100030E010006070200060702000E0384000E03C4000E01C8000E00F0020E00700 20700780403009C0401830E18007C03E001B1F7E9D20>38 D<004000800100020006000C 000C0018001800300030007000600060006000E000E000E000E000E000E000E000E000E0 00E000E000E000600060006000700030003000180018000C000C00060002000100008000 400A2A7D9E10>40 D<800040002000100018000C000C0006000600030003000380018001 80018001C001C001C001C001C001C001C001C001C001C001C001C0018001800180038003 000300060006000C000C00180010002000400080000A2A7E9E10>I<60F0F07010101010 20204080040C7C830C>44 DI<60F0F06004047C830C>I<000100 03000600060006000C000C000C0018001800180030003000300060006000C000C000C001 8001800180030003000300060006000C000C000C00180018001800300030003000600060 006000C000C00010297E9E15>I<03C00C301818300C300C700E60066006E007E007E007 E007E007E007E007E007E007E007E007E007E00760066006700E300C300C18180C3007E0 101D7E9B15>I<030007003F00C700070007000700070007000700070007000700070007 00070007000700070007000700070007000700070007000F80FFF80D1C7C9B15>I<07C0 1830201C400C400EF00FF80FF807F8077007000F000E000E001C001C00380070006000C0 0180030006010C01180110023FFE7FFEFFFE101C7E9B15>I<07E01830201C201C781E78 0E781E381E001C001C00180030006007E00030001C001C000E000F000F700FF80FF80FF8 0FF00E401C201C183007E0101D7E9B15>I<000C00000C00001C00003C00003C00005C00 00DC00009C00011C00031C00021C00041C000C1C00081C00101C00301C00201C00401C00 C01C00FFFFC0001C00001C00001C00001C00001C00001C00001C0001FFC0121C7F9B15> I<300C3FF83FF03FC020002000200020002000200023E024302818301C200E000E000F00 0F000F600FF00FF00FF00F800E401E401C2038187007C0101D7E9B15>I<00F0030C0604 0C0E181E301E300C700070006000E3E0E430E818F00CF00EE006E007E007E007E007E007 600760077006300E300C18180C3003E0101D7E9B15>I<4000007FFF807FFF007FFF0040 020080040080040080080000100000100000200000600000400000C00000C00001C00001 800001800003800003800003800003800007800007800007800007800007800007800003 0000111D7E9B15>I<03E00C301008200C20066006600660067006780C3E083FB01FE007 F007F818FC307E601E600FC007C003C003C003C00360026004300C1C1007E0101D7E9B15 >I<03C00C301818300C700C600EE006E006E007E007E007E007E0076007700F300F1817 0C2707C700060006000E300C780C78187010203030C00F80101D7E9B15>I<60F0F06000 00000000000000000060F0F06004127C910C>I<000600000006000000060000000F0000 000F0000000F00000017800000178000001780000023C0000023C0000023C0000041E000 0041E0000041E0000080F0000080F0000180F8000100780001FFF80003007C0002003C00 02003C0006003E0004001E0004001E000C001F001E001F00FF80FFF01C1D7F9C1F>65 DI<001F808000E061800180198007000780 0E0003801C0003801C00018038000180780000807800008070000080F0000000F0000000 F0000000F0000000F0000000F0000000F0000000F0000000700000807800008078000080 380000801C0001001C0001000E000200070004000180080000E03000001FC000191E7E9C 1E>IIII<001F808000E0618001801980070007800E000380 1C0003801C00018038000180780000807800008070000080F0000000F0000000F0000000 F0000000F0000000F0000000F000FFF0F0000F8070000780780007807800078038000780 1C0007801C0007800E00078007000B800180118000E06080001F80001C1E7E9C21>III<1FFF00F800780078007800780078007800780078007800780078007800780078007800 780078007800787078F878F878F878F0F040E021C01F00101D7F9B15>IIIII<003F800000E0E0000380380007001C000E000E001C0007003C000780380003807800 03C0780003C0700001C0F00001E0F00001E0F00001E0F00001E0F00001E0F00001E0F000 01E0F00001E0700001C0780003C0780003C0380003803C0007801C0007000E000E000700 1C000380380000E0E000003F80001B1E7E9C20>II82 D<07E0801C1980300580700380600180E0 0180E00080E00080E00080F00000F800007C00007FC0003FF8001FFE0007FF0000FF8000 0F800007C00003C00001C08001C08001C08001C0C00180C00180E00300D00200CC0C0083 F800121E7E9C17>I<7FFFFFC0700F01C0600F00C0400F0040400F0040C00F0020800F00 20800F0020800F0020000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000001F800003FFFC001B1C7F9B1E>IIII<7FF0FFC00FC03E000780180003C0 180003E0100001E0200001F0600000F0400000788000007D8000003D0000001E0000001F 0000000F0000000F8000000F80000013C0000023E0000021E0000041F00000C0F8000080 780001007C0003003C0002001E0006001F001F003F80FFC0FFF01C1C7F9B1F>II<7FFFF07C01F07001E06003C06003C0400780400F80400F00401E0000 1E00003C00007C0000780000F00000F00001E00003E00003C0100780100780100F00101F 00301E00203C00203C00607800E0F803E0FFFFE0141C7E9B19>I<080810102020404040 40808080808080B0B0F8F8787830300D0C7A9C15>92 D<1FC000307000783800781C0030 1C00001C00001C0001FC000F1C00381C00701C00601C00E01C40E01C40E01C40603C4030 4E801F870012127E9115>97 DI<07 E00C301878307870306000E000E000E000E000E000E00060007004300418080C3007C00E 127E9112>I<003F00000700000700000700000700000700000700000700000700000700 00070003E7000C1700180F00300700700700600700E00700E00700E00700E00700E00700 E00700600700700700300700180F000C370007C7E0131D7E9C17>I<03E00C301818300C 700E6006E006FFFEE000E000E000E00060007002300218040C1803E00F127F9112>I<00 F8018C071E061E0E0C0E000E000E000E000E000E00FFE00E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E007FE00F1D809C0D>I<00038003C4C00C38 C01C3880181800381C00381C00381C00381C001818001C38000C300013C0001000003000 001800001FF8001FFF001FFF803003806001C0C000C0C000C0C000C06001803003001C0E 0007F800121C7F9215>II<18003C 003C0018000000000000000000000000000000FC001C001C001C001C001C001C001C001C 001C001C001C001C001C001C001C001C00FF80091D7F9C0C>I<00C001E001E000C00000 0000000000000000000000000FE000E000E000E000E000E000E000E000E000E000E000E0 00E000E000E000E000E000E000E000E000E060E0F0C0F1C061803E000B25839C0D>IIIII<03F0000E1C001806 00300300700380600180E001C0E001C0E001C0E001C0E001C0E001C06001807003803003 001806000E1C0003F00012127F9115>II114 D<1F9030704030C010C010E010F8007F803FE00FF000F880388018C018C018E010D0 608FC00D127F9110>I<04000400040004000C000C001C003C00FFE01C001C001C001C00 1C001C001C001C001C001C101C101C101C101C100C100E2003C00C1A7F9910>IIII<7F8FF00F03800F030007020003840001C800 01D80000F00000700000780000F800009C00010E00020E000607000403801E07C0FF0FF8 1512809116>II<7FFC70386038407040F040E041C003C0 038007000F040E041C043C0C380870087038FFF80E127F9112>I E /Ft 7 117 df<00038000000380000007C0000007C0000007C000000FE000000FE000 001FF000001BF000001BF0000031F8000031F8000061FC000060FC0000E0FE0000C07E00 00C07E0001803F0001FFFF0003FFFF8003001F8003001F8006000FC006000FC00E000FE0 0C0007E0FFC07FFEFFC07FFE1F1C7E9B24>65 D<0FF8001C1E003E0F803E07803E07C01C 07C00007C0007FC007E7C01F07C03C07C07C07C0F807C0F807C0F807C0780BC03E13F80F E1F815127F9117>97 DI<03FC000E 0E001C1F003C1F00781F00780E00F80000F80000F80000F80000F80000F8000078000078 01803C01801C03000E0E0003F80011127E9115>I114 D<1FD830786018E018E018F000FF807FE07FF01FF807FC007C C01CC01CE01CE018F830CFC00E127E9113>I<0300030003000300070007000F000F003F FCFFFC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F08079803F00E 1A7F9913>I E /Fu 33 122 df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v 25 121 df<0000007C0000000000007C000000000000FE000000000000FE0000 00000000FE000000000001FF000000000001FF000000000003FF800000000003FF800000 000007FFC00000000007FFC00000000007FFC0000000000FFFE0000000000F7FE0000000 001F7FF0000000001E3FF0000000001E3FF0000000003E3FF8000000003C1FF800000000 7C1FFC00000000780FFC00000000780FFC00000000F80FFE00000000F007FE00000001F0 07FF00000001E003FF00000001E003FF00000003E003FF80000003C001FF80000007C001 FFC00000078000FFC00000078000FFC000000FFFFFFFE000000FFFFFFFE000001FFFFFFF F000001E00003FF000001E00003FF000003C00003FF800003C00001FF800007C00001FFC 00007800000FFC00007800000FFC0000F0000007FE0000F0000007FE0001F0000007FF00 03F8000003FF00FFFFC001FFFFFEFFFFC001FFFFFEFFFFC001FFFFFE37317DB03E>65 D<000003FF80018000003FFFF003800001FFFFFC0F800007FF007F1F80001FF8000FBF80 003FE00003FF8000FF800000FF8001FF0000007F8003FE0000003F8007FC0000003F8007 FC0000001F800FF80000001F801FF80000000F801FF00000000F803FF000000007803FF0 00000007807FF000000007807FE000000007807FE000000000007FE00000000000FFE000 00000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000 000000FFE00000000000FFE00000000000FFE000000000007FE000000000007FE0000000 00007FE000000000007FF000000003803FF000000003803FF000000003801FF000000003 801FF800000007800FF8000000070007FC000000070007FC0000000E0003FE0000001E00 01FF0000003C0000FF8000007800003FE00000F000001FF80003E0000007FF003F800000 01FFFFFE000000003FFFF80000000003FF80000031317BB03C>67 D69 D<000003FF80018000003FFFF003800001FFFFFC0F800007FF007F1F80001FF8000FBF80 003FE00003FF8000FF800000FF8001FF0000007F8003FE0000003F8007FC0000003F8007 FC0000001F800FF80000001F801FF80000000F801FF00000000F803FF000000007803FF0 00000007807FF000000007807FE000000007807FE000000000007FE00000000000FFE000 00000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000 000000FFE00000000000FFE00000000000FFE00007FFFFFE7FE00007FFFFFE7FE00007FF FFFE7FE0000001FF807FF0000001FF803FF0000001FF803FF0000001FF801FF0000001FF 801FF8000001FF800FF8000001FF8007FC000001FF8007FC000001FF8003FE000001FF80 01FF000001FF8000FF800001FF80003FE00003FF80001FF80007FF800007FF803F3F8000 01FFFFFE1F8000003FFFF80780000003FFC0018037317BB041>71 D77 D<00000FFE0000000000FFFFE000000007FFFFFC0000001FFC07FF0000003FE000FF8000 00FF80003FE00001FF00001FF00003FE00000FF80007FC000007FC0007FC000007FC000F F8000003FE001FF8000003FF001FF0000001FF003FF0000001FF803FF0000001FF803FF0 000001FF807FE0000000FFC07FE0000000FFC07FE0000000FFC0FFE0000000FFE0FFE000 0000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE00000 00FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE0FFE0000000FFE07FE0000000 FFC07FE0000000FFC07FF0000001FFC07FF0000001FFC03FF0000001FF803FF0000001FF 801FF8000003FF001FF8000003FF000FFC000007FE000FFC000007FE0007FE00000FFC00 03FF00001FF80001FF80003FF00000FFC0007FE000003FE000FF8000001FFC07FF000000 07FFFFFC00000000FFFFE0000000000FFE00000033317BB03E>79 DI<007FF8000003FFFF000007FFFFC0000FE01FE0001FF0 07F0001FF003F8001FF003FC001FF001FE000FE001FE0007C001FE00010001FE00000001 FE00000001FE000001FFFE00003FFFFE0001FFF1FE0007FE01FE000FF001FE001FC001FE 003F8001FE007F8001FE00FF0001FE00FF0001FE00FF0001FE00FF0001FE00FF0003FE00 7F8003FE007FC00EFE003FF03CFF000FFFF87FF807FFF03FF800FF800FF825207E9F28> 97 D<0007FF00007FFFE000FFFFF003FC03F807F007FC0FE007FC1FE007FC3FC007FC3F C003F87FC001F07F8000407F800000FF800000FF800000FF800000FF800000FF800000FF 800000FF800000FF8000007F8000007FC000007FC000003FC0000E3FE0000E1FE0001C0F F0001C07F8007803FF01F000FFFFE0007FFF800007FC001F207D9F25>99 D<0007FC0000003FFF800000FFFFE00003FC07F00007F801F8000FE000FC001FE0007E00 3FC0007E003FC0003F007FC0003F007F80003F007F80003F80FF80003F80FF80003F80FF FFFFFF80FFFFFFFF80FFFFFFFF80FF80000000FF80000000FF800000007F800000007F80 0000003FC00000003FC00003801FC00003801FE00007800FF0000F0007F8001E0003FE00 FC0000FFFFF800003FFFE0000003FF000021207E9F26>101 D<0000FF000007FFC0001F FFE0003FC7F0007F0FF800FE0FF801FE0FF801FC0FF803FC07F003FC03E003FC01C003FC 000003FC000003FC000003FC000003FC000003FC000003FC0000FFFFF800FFFFF800FFFF F80003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC00007FFF F0007FFFF0007FFFF0001D327EB119>I<001FF007E000FFFE3FF001FFFF7FF807F83FF1 F80FE00FE1F80FE00FE0F01FC007F0601FC007F0003FC007F8003FC007F8003FC007F800 3FC007F8003FC007F8001FC007F0001FC007F0000FE00FE0000FE00FE00007F83FC00007 FFFF000006FFFE00000E1FF000000E000000001E000000001E000000001F000000001F80 0000001FFFFFC0000FFFFFF8000FFFFFFE0007FFFFFF0003FFFFFF8007FFFFFFC01FFFFF FFE03F00007FE07E00000FF0FC000007F0FC000003F0FC000003F0FC000003F0FC000003 F07E000007E03F00000FC01FC0003F800FF801FF0007FFFFFE0000FFFFF000001FFF8000 252F7E9F29>I<01F800000000FFF800000000FFF800000000FFF8000000000FF8000000 0007F80000000007F80000000007F80000000007F80000000007F80000000007F8000000 0007F80000000007F80000000007F80000000007F80000000007F80000000007F8000000 0007F80000000007F807F8000007F83FFF000007F87FFF800007F8F03FC00007F9C01FE0 0007FB000FE00007FE000FF00007FE000FF00007FC000FF00007FC000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 0007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF0 00FFFFC1FFFF80FFFFC1FFFF80FFFFC1FFFF8029327DB12E>I<03C0000FF0000FF0001F F8001FF8001FFC001FF8001FF8000FF0000FF00003C00000000000000000000000000000 000000000000000000000001F800FFF800FFF800FFF8000FF80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F800FFFF80FFFF80FFFF8011 337DB217>I<01F800FFF800FFF800FFF8000FF80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F800FFFFC0FFFFC0FFFFC012327DB117>108 D<03F007F8000FF000FFF03FFF007FFE 00FFF07FFF80FFFF00FFF0F03FC1E07F800FF1C01FE3803FC007F3000FE6001FC007F600 0FFC001FE007FE000FFC001FE007FC000FF8001FE007FC000FF8001FE007F8000FF0001F E007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F800 0FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001F E007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F800 0FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001F E0FFFFC1FFFF83FFFFFFFFC1FFFF83FFFFFFFFC1FFFF83FFFF40207D9F45>I<03F007F8 0000FFF03FFF0000FFF07FFF8000FFF0F03FC0000FF1C01FE00007F3000FE00007F6000F F00007FE000FF00007FC000FF00007FC000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FFFFC1FFFF80FFFFC1FF FF80FFFFC1FFFF8029207D9F2E>I<0007FE0000003FFFC00000FFFFF00003FC03FC0007 F000FE000FE0007F001FC0003F803FC0003FC03FC0003FC07F80001FE07F80001FE07F80 001FE0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF8000 1FF0FF80001FF07F80001FE07F80001FE07F80001FE03FC0003FC03FC0003FC01FE0007F 800FE0007F0007F801FE0003FE07FC0001FFFFF800003FFFC0000007FE000024207E9F29 >I<01F80FF000FFF87FFE00FFF9FFFF80FFFFE07FC00FFF001FE007FE000FF007F80007 F807F80007FC07F80003FC07F80003FE07F80003FE07F80001FE07F80001FF07F80001FF 07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FE07 F80003FE07F80003FE07F80003FC07F80007FC07FC0007F807FE000FF007FF001FE007FB E07FC007F9FFFF0007F87FFE0007F81FE00007F800000007F800000007F800000007F800 000007F800000007F800000007F800000007F800000007F800000007F800000007F80000 00FFFFC00000FFFFC00000FFFFC00000282E7E9F2E>I<03F03F00FFF07FC0FFF1FFE0FF F3C7F00FF38FF807F70FF807F60FF807FE0FF807FC07F007FC03E007FC008007F8000007 F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007 F8000007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FF FFE000FFFFE0001D207E9F22>114 D<00FF870007FFEF001FFFFF003F007F003C001F00 78000F00F8000700F8000700F8000700FC000700FF000000FFF800007FFFC0003FFFF000 3FFFFC000FFFFE0007FFFF0001FFFF80001FFF800000FFC000001FC060000FC0E00007C0 E00007C0F00007C0F8000780F8000F80FE000F00FF803E00FFFFFC00F3FFF800C07FC000 1A207D9F21>I<00380000380000380000380000380000780000780000780000F80000F8 0001F80003F80007F8001FF800FFFFFEFFFFFEFFFFFE07F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0707F80707F80707F80707F80707F80707F80703F80E03FC0E01FE1C00FFF8007FF0000F E0182E7EAD20>I<01F80003F000FFF801FFF000FFF801FFF000FFF801FFF0000FF8001F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8001FF00007F8001FF00003F8003FF00003F8006FF00001FE03CF F80000FFFF8FFF80007FFF0FFF80000FFC0FFF8029207D9F2E>II120 D E /Fw 85 125 df<00008000000001C000000001C000000003E000000003E00000 0005F000000004F000000008F80000000878000000107C000000103C000000203E000000 201E000000401F000000400F000000800F80000080078000010007C000010003C0000200 03E000020001E000040001F000040000F000080000F80008000078001000007C00100000 3C002000003E002000001E007FFFFFFF007FFFFFFF00FFFFFFFF8021207E9F26>1 D<001F800000F0F00001C0380007801E000F000F000E0007001E0007803C0003C03C0003 C07C0003E07C0003E0780001E0F80001F0F84021F0F84021F0F87FE1F0F87FE1F0F87FE1 F0F84021F0F84021F0F80001F0780001E0780001E07C0003E03C0003C03C0003C01E0007 800E0007000F000F0007801E0001C0380000F0F000001F80001C217D9F23>I5 DI<001F83E000F06E3001C078780380F8780300F030070070000700700007 00700007007000070070000700700007007000FFFFFF8007007000070070000700700007 007000070070000700700007007000070070000700700007007000070070000700700007 00700007007000070070000700700007007000070070007FE3FF001D20809F1B>11 D<003F0000E0C001C0C00381E00701E00701E00700000700000700000700000700000700 00FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700 E00700E00700E00700E00700E00700E00700E00700E07FC3FE1720809F19>I<003FE000 E0E001C1E00381E00700E00700E00700E00700E00700E00700E00700E00700E0FFFFE007 00E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E007 00E00700E00700E00700E00700E00700E07FE7FE1720809F19>I<001F81F80000F04F04 0001C07C06000380F80F000300F00F000700F00F00070070000007007000000700700000 070070000007007000000700700000FFFFFFFF0007007007000700700700070070070007 007007000700700700070070070007007007000700700700070070070007007007000700 700700070070070007007007000700700700070070070007007007000700700700070070 07007FE3FE3FF02420809F26>I<7038F87CFC7EFC7E743A040204020402080408041008 1008201040200F0E7E9F17>34 D<00780000008400000184000003020000070200000702 00000702000007020000070400000704000007080000070800000310000003A00FFC03C0 03E0038001C001C0008001C0010003E0010004E0020008F0020018700400307808007038 0800701C1000F01E1000F00E2000F0074000F003C0087003C0087801C010380670301C18 386007E00F801E227EA023>38 D<70F8FCFC74040404080810102040060E7C9F0D>I<00 20004000800100020006000C000C00180018003000300030007000600060006000E000E0 00E000E000E000E000E000E000E000E000E000E000600060006000700030003000300018 0018000C000C000600020001000080004000200B2E7DA112>I<80004000200010000800 0C00060006000300030001800180018001C000C000C000C000E000E000E000E000E000E0 00E000E000E000E000E000E000C000C000C001C001800180018003000300060006000C00 080010002000400080000B2E7DA112>I<00060000000600000006000000060000000600 000006000000060000000600000006000000060000000600000006000000060000000600 0000060000FFFFFFF0FFFFFFF00006000000060000000600000006000000060000000600 000006000000060000000600000006000000060000000600000006000000060000000600 001C207D9A23>43 D<70F8FCFC74040404080810102040060E7C840D>II<70F8F8F87005057C840D>I<03F0000E1C001C0E001806003807007003807003 80700380700380F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003 C0F003C0F003C0F003C07003807003807003807807803807001806001C0E000E1C0003F0 00121F7E9D17>48 D<018003800F80F38003800380038003800380038003800380038003 800380038003800380038003800380038003800380038003800380038007C0FFFE0F1E7C 9D17>I<03F0000C1C00100E00200700400780800780F007C0F803C0F803C0F803C02007 C00007C0000780000780000F00000E00001C0000380000700000600000C0000180000300 000600400C00401800401000803FFF807FFF80FFFF80121E7E9D17>I<03F0000C1C0010 0E00200F00780F80780780780780380F80000F80000F00000F00000E00001C0000380003 F000003C00000E00000F000007800007800007C02007C0F807C0F807C0F807C0F0078040 0780400F00200E001C3C0003F000121F7E9D17>I<000600000600000E00000E00001E00 002E00002E00004E00008E00008E00010E00020E00020E00040E00080E00080E00100E00 200E00200E00400E00C00E00FFFFF0000E00000E00000E00000E00000E00000E00000E00 00FFE0141E7F9D17>I<1803001FFE001FFC001FF8001FE0001000001000001000001000 0010000010000011F000161C00180E001007001007800003800003800003C00003C00003 C07003C0F003C0F003C0E00380400380400700200600100E000C380003E000121F7E9D17 >I<007C000182000701000E03800C07801C0780380300380000780000700000700000F1 F000F21C00F40600F80700F80380F80380F003C0F003C0F003C0F003C0F003C07003C070 03C07003803803803807001807000C0E00061C0001F000121F7E9D17>I<4000007FFFC0 7FFF807FFF80400100800200800200800400000800000800001000002000002000004000 00400000C00000C00001C000018000038000038000038000038000078000078000078000 078000078000078000078000030000121F7D9D17>I<03F0000C0C001006003003002001 806001806001806001807001807803003E03003F06001FC8000FF00003F80007FC000C7E 00103F00300F806003804001C0C001C0C000C0C000C0C000C0C000806001802001001002 000C0C0003F000121F7E9D17>I<03F0000E18001C0C00380600380700700700700380F0 0380F00380F003C0F003C0F003C0F003C0F003C07007C07007C03807C0180BC00E13C003 E3C0000380000380000380000700300700780600780E00700C002018001070000FC00012 1F7E9D17>I<70F8F8F8700000000000000000000070F8F8F87005147C930D>I<70F8F8F8 700000000000000000000070F0F8F878080808101010202040051D7C930D>I<7FFFFFE0 FFFFFFF00000000000000000000000000000000000000000000000000000000000000000 FFFFFFF07FFFFFE01C0C7D9023>61 D<000100000003800000038000000380000007C000 0007C0000007C0000009E0000009E0000009E0000010F0000010F0000010F00000207800 002078000020780000403C0000403C0000403C0000801E0000801E0000FFFE0001000F00 01000F0001000F00020007800200078002000780040003C00E0003C01F0007E0FFC03FFE 1F207F9F22>65 DI<000FC04000 7030C001C009C0038005C0070003C00E0001C01E0000C01C0000C03C0000C07C0000407C 00004078000040F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8 000000F8000000780000007C0000407C0000403C0000401C0000401E0000800E00008007 0001000380020001C0040000703800000FC0001A217D9F21>III< FFFFFF000F800F0007800300078003000780010007800180078000800780008007800080 07800080078080000780800007808000078080000781800007FF80000781800007808000 078080000780800007808000078000000780000007800000078000000780000007800000 07800000078000000FC00000FFFE0000191F7E9E1E>I<000FE0200078186000E004E003 8002E0070001E00F0000E01E0000601E0000603C0000603C0000207C00002078000020F8 000000F8000000F8000000F8000000F8000000F8000000F8000000F8007FFCF80003E078 0001E07C0001E03C0001E03C0001E01E0001E01E0001E00F0001E0070001E0038002E000 E0046000781820000FE0001E217D9F24>III<0FFFC0007C00003C00003C00003C00003C00003C00003C00003C00003C00003C0000 3C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C0020 3C00F83C00F83C00F83C00F0380040780040700030E0000F800012207E9E17>I76 DII<001F800000 F0F00001C0380007801E000F000F000E0007001E0007803C0003C03C0003C07C0003E078 0001E0780001E0F80001F0F80001F0F80001F0F80001F0F80001F0F80001F0F80001F0F8 0001F0F80001F0780001E07C0003E07C0003E03C0003C03C0003C01E0007800E0007000F 000F0007801E0001C0380000F0F000001F80001C217D9F23>II<001F800000F0F00001C0380007801E000F000F000E0007 001E0007803C0003C03C0003C07C0003E07C0003E0780001E0F80001F0F80001F0F80001 F0F80001F0F80001F0F80001F0F80001F0F80001F0F80001F0780001E0780001E07C0003 E03C0003C03C0F03C01E1087800E20C7000F20CF0007A0FE0001E0F80000F0F010001FF0 1000007010000078300000387000003FF000003FE000001FE000000FC0000007801C297D 9F23>II<07E0800C19 80100780300380600180600180E00180E00080E00080E00080F00000F000007800007F00 003FF0001FFC000FFE0003FF00001F800007800003C00003C00001C08001C08001C08001 C08001C0C00180C00380E00300F00600CE0C0081F80012217D9F19>I<7FFFFFE0780F01 E0600F0060400F0020400F0020C00F0030800F0010800F0010800F0010800F0010000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00001F800007FFFE001C1F7E9E21>II87 D<7FF83FF80FE00FC007 C0070003C0020001E0040001F00C0000F0080000781000007C1000003C2000003E400000 1E4000000F8000000F8000000780000003C0000007E0000005E0000009F0000018F80000 10780000207C0000603C0000401E0000801F0001800F0001000780020007C0070003C01F 8007E0FFE01FFE1F1F7F9E22>II91 D<080410082010201040204020804080408040 B85CFC7EFC7E7C3E381C0F0E7B9F17>II<081020 204040808080B8FCFC7C38060E7D9F0D>96 D<1FE000303000781800781C00300E00000E 00000E00000E0000FE00078E001E0E00380E00780E00F00E10F00E10F00E10F01E10781E 103867200F83C014147E9317>I<0E0000FE00000E00000E00000E00000E00000E00000E 00000E00000E00000E00000E00000E3E000EC3800F01C00F00E00E00E00E00700E00700E 00780E00780E00780E00780E00780E00780E00700E00700E00E00F00E00D01C00CC30008 3E0015207F9F19>I<03F80E0C1C1E381E380C70007000F000F000F000F000F000F00070 007000380138011C020E0C03F010147E9314>I<000380003F8000038000038000038000 038000038000038000038000038000038000038003E380061B801C078038038038038070 0380700380F00380F00380F00380F00380F00380F003807003807003803803803807801C 07800E1B8003E3F815207E9F19>I<03F0000E1C001C0E00380700380700700700700380 F00380F00380FFFF80F00000F00000F000007000007000003800801800800C0100070600 01F80011147F9314>I<007C00C6018F038F07060700070007000700070007000700FFF0 070007000700070007000700070007000700070007000700070007000700070007000700 7FF01020809F0E>I<0000E003E3300E3C301C1C30380E00780F00780F00780F00780F00 780F00380E001C1C001E380033E0002000002000003000003000003FFE001FFF800FFFC0 3001E0600070C00030C00030C00030C000306000603000C01C038003FC00141F7F9417> I<0E0000FE00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00 000E3E000E43000E81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0FFE7FC16207F9F19>I<1C003E00 3E003E001C000000000000000000000000000E007E000E000E000E000E000E000E000E00 0E000E000E000E000E000E000E000E000E000E00FFC00A1F809E0C>I<00E001F001F001 F000E0000000000000000000000000007007F000F0007000700070007000700070007000 7000700070007000700070007000700070007000700070007000706070F060F0C061803F 000C28829E0E>I<0E0000FE00000E00000E00000E00000E00000E00000E00000E00000E 00000E00000E00000E0FF00E03C00E03000E02000E04000E08000E10000E30000E70000E F8000F38000E1C000E1E000E0E000E07000E07800E03800E03C00E03E0FFCFF815207F9F 18>I<0E00FE000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E000E000E000E000E00FFE00B20809F0C >I<0E1F01F000FE618618000E81C81C000F00F00E000F00F00E000E00E00E000E00E00E 000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E00 0E00E00E000E00E00E000E00E00E000E00E00E000E00E00E00FFE7FE7FE023147F9326> I<0E3E00FE43000E81800F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0FFE7FC16147F9319>I<01F80007 0E001C03803801C03801C07000E07000E0F000F0F000F0F000F0F000F0F000F0F000F070 00E07000E03801C03801C01C0380070E0001F80014147F9317>I<0E3E00FEC3800F01C0 0F00E00E00E00E00F00E00700E00780E00780E00780E00780E00780E00780E00700E00F0 0E00E00F01E00F01C00EC3000E3E000E00000E00000E00000E00000E00000E00000E0000 0E0000FFE000151D7F9319>I<03E0800619801C05803C0780380380780380700380F003 80F00380F00380F00380F00380F003807003807803803803803807801C0B800E138003E3 80000380000380000380000380000380000380000380000380003FF8151D7E9318>I<0E 78FE8C0F1E0F1E0F0C0E000E000E000E000E000E000E000E000E000E000E000E000E000E 00FFE00F147F9312>I<1F9030704030C010C010C010E00078007F803FE00FF000708038 80188018C018C018E030D0608F800D147E9312>I<020002000200060006000E000E003E 00FFF80E000E000E000E000E000E000E000E000E000E000E000E080E080E080E080E0806 10031001E00D1C7F9B12>I<0E01C0FE1FC00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E03C00603C0030DC001F1FC 16147F9319>III<7FC3FC0F01E00701C007018003810001C20000 E40000EC00007800003800003C00007C00004E000087000107000303800201C00601E01E 01E0FF07FE1714809318>II<3FFF 380E200E201C40384078407000E001E001C00380078007010E011E011C0338027006700E FFFE10147F9314>III E /Fx 51 123 df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end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 0 195 a Fx(Neur)n(al)16 b(Computation)p Fw(,)g Fx(8)p Fw(,)f(129{151,)d(1996.)14 385 y Fv(On)28 b(Con)n(v)n(ergence)d (Prop)r(erties)i(of)h(the)f(EM)h(Algorithm)g(for)635 454 y(Gaussian)e(Mixtures)902 591 y Fu(Lei)16 b(Xu)618 662 y(Departmen)o(t)f(of)i(Computer)e(Science)567 731 y(The)h(Chinese)g(Univ)o(ersit)o(y)e(of)i(Hong)h(Kong)786 851 y(Mic)o(hael)e(I.)g(Jordan)499 922 y(Departmen)o(t)g(of)i(Brain)e (and)i(Cognitiv)o(e)f(Sciences)567 991 y(Massac)o(h)o(usetts)h (Institute)e(of)i(T)l(ec)o(hnology)883 1201 y Ft(Abstract)176 1288 y Fs(W)m(e)e(build)g(up)g(the)h(mathematical)c(connection)k(b)q (et)o(w)o(een)h(the)f(\\Exp)q(ectation-Maximization")d(\(EM\))114 1349 y(algorithm)h(and)k(gradien)o(t-based)g(approac)o(hes)g(for)f (maxim)n(um)c(lik)o(eliho)q(o)q(d)j(learning)h(of)g(\014nite)h (Gaussian)114 1410 y(mixtures.)23 b(W)m(e)16 b(sho)o(w)g(that)g(the)h (EM)f(step)h(in)e(parameter)h(space)h(is)f(obtained)g(from)e(the)i (gradien)o(t)g(via)f(a)114 1470 y(pro)r(jection)d(matrix)f Fr(P)6 b Fs(,)12 b(and)g(w)o(e)g(pro)o(vide)h(an)f(explicit)g (expression)h(for)f(the)h(matrix.)j(W)m(e)c(then)h(analyze)g(the)114 1531 y(con)o(v)o(ergence)18 b(of)e(EM)h(in)f(terms)g(of)g(sp)q(ecial)h (prop)q(erties)h(of)e Fr(P)22 b Fs(and)16 b(pro)o(vide)h(new)g(results) g(analyzing)f(the)114 1592 y(e\013ect)i(that)f Fr(P)22 b Fs(has)17 b(on)f(the)h(lik)o(eliho)q(o)q(d)e(surface.)28 b(Based)18 b(on)e(these)i(mathematical)c(results,)k(w)o(e)f(presen)o(t) 114 1653 y(a)d(comparativ)o(e)f(discussion)i(of)f(the)i(adv)n(an)o (tages)e(and)g(disadv)n(an)o(tages)g(of)g(EM)h(and)g(other)g (algorithms)d(for)114 1713 y(the)i(learning)f(of)h(Gaussian)f(mixture)g (mo)q(dels.)0 1873 y Fq(1)69 b(In)n(tro)r(duction)0 1987 y Fw(The)17 b(\\Exp)q(ectation-Maximization")h(\(EM\))e(algorithm)h(is) g(a)g(general)g(tec)o(hnique)i(for)d(maxim)o(um)h(lik)o(eliho)q(o)q(d)0 2056 y(\(ML\))i(or)h(maxim)o(um)f(a)h(p)q(osteriori)g(\(MAP\))f (estimation.)34 b(The)20 b(recen)o(t)g(emphasis)h(in)f(the)g(neural)h (net)o(w)o(ork)0 2125 y(literature)h(on)f(probabilistic)i(mo)q(dels)f (has)g(led)g(to)e(increased)j(in)o(terest)e(in)h(EM)f(as)g(a)g(p)q (ossible)i(alternativ)o(e)0 2194 y(to)17 b(gradien)o(t-based)i(metho)q (ds)f(for)g(optimization.)29 b(EM)18 b(has)g(b)q(een)h(used)g(for)e(v)m (ariations)i(on)f(the)g(traditional)0 2262 y(theme)e(of)g(Gaussian)g (mixture)h(mo)q(deling)h(\(Ghahramani)d(&)i(Jordan,)f(1994;)f(No)o (wlan,)h(1991;)f(Xu)i(&)f(Jordan,)0 2331 y(1993a,)h(b;)i(T)l(resp,)f (Ahmad)g(&)g(Neuneier,)i(1994;)e(Xu,)g(Jordan)g(&)h(Hin)o(ton,)f (1994\))e(and)j(has)e(also)h(b)q(een)h(used)0 2400 y(for)e(no)o(v)o(el) i(c)o(hain-structured)g(and)f(tree-structured)g(arc)o(hitectures)h (\(Bengio)f(&)h(F)l(rasconi,)f(1995;)g(Jordan)g(&)0 2469 y(Jacobs,)h(1994\).)27 b(The)18 b(empirical)i(results)f(rep)q(orted)f (in)h(these)f(pap)q(ers)h(suggest)e(that)g(EM)h(has)g(considerable)0 2538 y(promise)e(as)f(an)h(optimization)g(metho)q(d)g(for)f(suc)o(h)h (arc)o(hitectures.)21 b(Moreo)o(v)o(er,)14 b(new)i(theoretical)g (results)g(ha)o(v)o(e)0 2607 y(b)q(een)h(obtained)h(that)d(link)j(EM)e (to)g(other)g(topics)g(in)i(learning)f(theory)f(\(Amari,)g(1994;)g (Jordan)g(&)h(Xu,)f(1993;)0 2676 y(Neal)g(&)f(Hin)o(ton,)g(1993;)f(Xu)i (&)f(Jordan,)g(1993c;)f(Y)l(uille,)j(Stolorz)f(&)f(Utans,)f(1994\).)963 2817 y(1)p eop %%Page: 2 2 2 1 bop 71 195 a Fw(Despite)16 b(these)h(dev)o(elopmen)o(ts,)g(there)f (are)g(grounds)h(for)e(caution)i(ab)q(out)f(the)h(promise)g(of)f(the)g (EM)g(algo-)0 264 y(rithm.)30 b(One)20 b(reason)e(for)g(caution)h (comes)g(from)f(consideration)h(of)f(theoretical)i(con)o(v)o(ergence)e (rates,)h(whic)o(h)0 333 y(sho)o(w)f(that)g(EM)h(is)g(a)g(\014rst)f (order)h(algorithm.)826 316 y Fp(1)877 333 y Fw(More)f(precisely)l(,)j (there)e(are)g(t)o(w)o(o)e(k)o(ey)i(results)g(a)o(v)m(ailable)i(in)0 402 y(the)16 b(statistical)g(literature)h(on)f(the)g(con)o(v)o(ergence) g(of)g(EM.)f(First,)h(it)g(has)g(b)q(een)h(established)g(that)f(under)g (mild)0 471 y(conditions)h(EM)f(is)h(guaran)o(teed)f(to)f(con)o(v)o (erge)h(to)o(w)o(ard)f(a)h(lo)q(cal)h(maxim)o(um)f(of)g(the)g(log)g (lik)o(eliho)q(o)q(d)j Fo(l)e Fw(\(Bo)o(yles,)0 539 y(1983;)11 b(Dempster,)h(Laird)g(&)g(Rubin,)i(1977;)e(Redner)g(&)g(W)l(alk)o(er,)g (1984;)g(W)l(u,)g(1983\).)17 b(\(Indeed)c(the)f(con)o(v)o(ergence)0 608 y(is)17 b(monotonic:)24 b Fo(l)q Fw(\(\002)359 592 y Fp(\()p Fn(k)q Fp(+1\))452 608 y Fw(\))15 b Fm(\025)h Fo(l)q Fw(\(\002)604 592 y Fp(\()p Fn(k)q Fp(\))652 608 y Fw(\),)g(where)h(\002)867 592 y Fp(\()p Fn(k)q Fp(\))933 608 y Fw(is)h(the)f(v)m(alue)h(of)e(the)h(parameter)f(v)o(ector)g(\002) h(at)g(iteration)0 677 y Fo(k)q Fw(.\))28 b(Second,)19 b(considering)h(EM)e(as)f(a)h(mapping)h(\002)910 661 y Fp(\()p Fn(k)q Fp(+1\))1021 677 y Fw(=)f Fo(M)5 b Fw(\(\002)1176 661 y Fp(\()p Fn(k)q Fp(\))1225 677 y Fw(\))17 b(with)i(\014xed)g(p)q (oin)o(t)f(\002)1636 661 y Fl(\003)1674 677 y Fw(=)f Fo(M)5 b Fw(\(\002)1828 661 y Fl(\003)1848 677 y Fw(\),)18 b(w)o(e)0 746 y(ha)o(v)o(e)d(\002)140 730 y Fp(\()p Fn(k)q Fp(+1\))244 746 y Fm(\000)c Fw(\002)325 730 y Fl(\003)357 746 y Fm(\031)410 724 y Fn(@)r(M)t Fp(\(\002)510 712 y Fk(\003)527 724 y Fp(\))p 410 736 131 2 v 443 762 a Fn(@)r Fp(\002)491 752 y Fk(\003)546 746 y Fw(\(\002)599 730 y Fp(\()p Fn(k)q Fp(\))658 746 y Fm(\000)f Fw(\002)738 730 y Fl(\003)758 746 y Fw(\))15 b(when)h(\002)945 730 y Fp(\()p Fn(k)q Fp(+1\))1054 746 y Fw(is)g(near)f(\002)1236 730 y Fl(\003)1256 746 y Fw(,)g(and)g(th)o(us)548 876 y Fm(k)p Fw(\002)606 858 y Fp(\()p Fn(k)q Fp(+1\))710 876 y Fm(\000)10 b Fw(\002)790 858 y Fl(\003)810 876 y Fm(k)j(\024)g(k)922 846 y Fo(@)s(M)5 b Fw(\(\002)1051 829 y Fl(\003)1070 846 y Fw(\))p 922 866 167 2 v 964 908 a Fo(@)s Fw(\002)1026 894 y Fl(\003)1093 876 y Fm(k)k(\001)h(k)p Fw(\002)1206 858 y Fp(\()p Fn(k)q Fp(\))1265 876 y Fm(\000)h Fw(\002)1346 858 y Fl(\003)1366 876 y Fm(k)p Fo(;)0 991 y Fw(with)822 1060 y Fm(k)850 1029 y Fo(@)s(M)5 b Fw(\(\002)979 1013 y Fl(\003)998 1029 y Fw(\))p 850 1049 V 892 1091 a Fo(@)s Fw(\002)954 1078 y Fl(\003)1021 1060 y Fm(k)12 b(6)p Fw(=)h(0)0 1156 y(almost)i(surely)l(.)21 b(That)14 b(is,)i(EM)e(is)i(a)f(\014rst)g(order)g(algorithm.)71 1225 y(The)g(\014rst-order)g(con)o(v)o(ergence)h(of)f(EM)g(has)g(b)q (een)i(cited)f(in)h(the)e(statistical)h(literature)g(as)f(a)g(ma)s(jor) f(dra)o(w-)0 1293 y(bac)o(k.)20 b(Redner)15 b(and)g(W)l(alk)o(er)f (\(1984\),)f(in)i(a)f(widely-cited)j(article,)e(argued)g(that)f(sup)q (erlinear)i(\(quasi-Newton,)0 1362 y(metho)q(d)d(of)f(scoring\))h(and)g (second-order)g(\(Newton\))f(metho)q(ds)h(should)g(generally)h(b)q(e)f (preferred)h(to)e(EM.)g(They)0 1431 y(rep)q(orted)h(empirical)h (results)f(demonstrating)g(the)f(slo)o(w)h(con)o(v)o(ergence)g(of)f(EM) g(on)h(a)f(Gaussian)h(mixture)g(mo)q(del)0 1500 y(problem)j(for)e(whic) o(h)i(the)f(mixture)g(comp)q(onen)o(ts)g(w)o(ere)f(not)h(w)o(ell)h (separated.)j(These)d(results)f(did)h(not)e(include)0 1569 y(tests)i(of)h(comp)q(eting)g(algorithms,)g(ho)o(w)o(ev)o(er.)24 b(Moreo)o(v)o(er,)16 b(ev)o(en)h(though)g(the)g(con)o(v)o(ergence)g(to) o(w)o(ard)e(the)i(\\op-)0 1638 y(timal")h(parameter)g(v)m(alues)h(w)o (as)e(slo)o(w)h(in)h(these)f(exp)q(erimen)o(ts,)h(the)f(con)o(v)o (ergence)h(in)f(lik)o(eliho)q(o)q(d)j(w)o(as)d(rapid.)0 1707 y(Indeed,)c(Redner)g(and)f(W)l(alk)o(er)f(ac)o(kno)o(wledge)h (that)f(their)h(results)g(sho)o(w)f(that)g(\\...)18 b(ev)o(en)13 b(when)g(the)g(comp)q(onen)o(t)0 1776 y(p)q(opulations)j(in)g(a)e (mixture)h(are)f(p)q(o)q(orly)i(separated,)e(the)h(EM)f(algorithm)h (can)g(b)q(e)g(exp)q(ected)h(to)e(pro)q(duce)i(in)f(a)0 1844 y(v)o(ery)c(small)i(n)o(um)o(b)q(er)e(of)h(iterations)f(parameter) g(v)m(alues)i(suc)o(h)f(that)f(the)g(mixture)h(densit)o(y)g(determined) h(b)o(y)f(them)0 1913 y(re\015ects)17 b(the)g(sample)g(data)f(v)o(ery)g (w)o(ell.")25 b(In)17 b(the)g(con)o(text)f(of)g(the)h(curren)o(t)f (literature)h(on)g(learning,)g(in)h(whic)o(h)0 1982 y(the)g(predictiv)o (e)i(asp)q(ect)e(of)f(data)h(mo)q(deling)h(is)f(emphasized)i(at)d(the)h (exp)q(ense)i(of)d(the)h(traditional)h(Fisherian)0 2051 y(statistician's)13 b(concern)g(o)o(v)o(er)f(the)h(\\true")f(v)m(alues) i(of)e(parameters,)g(suc)o(h)h(rapid)g(con)o(v)o(ergence)g(in)g(lik)o (eliho)q(o)q(d)j(is)d(a)0 2120 y(ma)s(jor)e(desideratum)j(of)e(a)g (learning)i(algorithm)e(and)h(undercuts)h(the)e(critique)i(of)f(EM)f (as)g(a)g(\\slo)o(w")g(algorithm.)71 2189 y(In)17 b(the)g(curren)o(t)f (pap)q(er,)h(w)o(e)g(pro)o(vide)g(a)g(comparativ)o(e)f(analysis)h(of)g (EM)f(and)h(other)f(optimization)i(meth-)0 2258 y(o)q(ds.)28 b(W)l(e)18 b(emphasize)h(the)f(comparison)g(b)q(et)o(w)o(een)g(EM)f (and)h(other)g(\014rst-order)f(metho)q(ds)h(\(gradien)o(t)f(ascen)o(t,) 0 2327 y(conjugate)d(gradien)o(t)h(metho)q(ds\),)f(b)q(ecause)i(these)f (ha)o(v)o(e)f(tended)h(to)f(b)q(e)i(the)e(metho)q(ds)h(of)f(c)o(hoice)i (in)f(the)g(neural)0 2395 y(net)o(w)o(ork)h(literature.)25 b(Ho)o(w)o(ev)o(er,)16 b(w)o(e)h(also)f(compare)h(EM)f(to)h(sup)q (erlinear)h(and)f(second-order)h(metho)q(ds.)25 b(W)l(e)0 2464 y(argue)14 b(that)g(EM)g(has)g(a)h(n)o(um)o(b)q(er)f(of)g(adv)m (an)o(tages,)g(including)j(its)e(naturalness)g(at)f(handling)i(the)e (probabilistic)0 2533 y(constrain)o(ts)k(of)g(mixture)g(problems)h(and) g(its)f(guaran)o(tees)g(of)g(con)o(v)o(ergence.)29 b(W)l(e)19 b(also)f(pro)o(vide)h(new)f(results)p 0 2573 780 2 v 52 2602 a Fj(1)69 2618 y Fi(An)c(iterativ)o(e)h(algorithm)h(is)e(said)h (to)f(ha)o(v)o(e)g(a)g(lo)q(cal)h(con)o(v)o(ergence)g(rate)f(of)f (order)h Fh(q)g Fg(\025)d Fi(1)j(if)g Fg(k)p Fi(\002)1469 2602 y Fj(\()p Ff(k)q Fj(+1\))1559 2618 y Fg(\000)9 b Fi(\002)1628 2602 y Fk(\003)1645 2618 y Fg(k)p Fh(=)p Fg(k)p Fi(\002)1732 2602 y Fj(\()p Ff(k)q Fj(\))1784 2618 y Fg(\000)g Fi(\002)1853 2602 y Fk(\003)1871 2618 y Fg(k)1890 2602 y Ff(q)1919 2618 y Fg(\024)0 2674 y Fh(r)h Fi(+)e Fh(o)p Fi(\()p Fg(k)p Fi(\002)148 2658 y Fj(\()p Ff(k)q Fj(\))199 2674 y Fg(\000)h Fi(\002)268 2658 y Fk(\003)286 2674 y Fg(k)p Fi(\))k(for)f Fh(k)i Fi(su\016cien)o(tly)i(large.)963 2817 y Fw(2)p eop %%Page: 3 3 3 2 bop 0 195 a Fw(suggesting)16 b(that)g(under)h(appropriate)f (conditions)h(EM)f(ma)o(y)g(in)h(fact)e(appro)o(ximate)h(a)g(sup)q (erlinear)i(metho)q(d;)0 264 y(this)j(w)o(ould)f(explain)i(some)e(of)g (the)h(promising)g(empirical)h(results)f(that)e(ha)o(v)o(e)h(b)q(een)h (obtained)g(\(Jordan)f(&)0 333 y(Jacobs,)d(1994\),)f(and)h(w)o(ould)h (further)f(temp)q(er)g(the)g(critique)i(of)d(EM)h(o\013ered)g(b)o(y)g (Redner)h(and)f(W)l(alk)o(er.)26 b(The)0 402 y(analysis)14 b(in)h(the)e(curren)o(t)h(pap)q(er)g(fo)q(cuses)g(on)f(unsup)q(ervised) j(learning;)f(for)e(related)h(results)g(in)g(the)g(sup)q(ervised)0 471 y(learning)i(domain)g(see)f(Jordan)h(and)f(Xu)h(\(in)f(press\).)71 539 y(The)d(remainder)h(of)f(the)h(pap)q(er)f(is)h(organized)g(as)f (follo)o(ws.)19 b(W)l(e)12 b(\014rst)g(brie\015y)i(review)f(the)f(EM)g (algorithm)g(for)0 608 y(Gaussian)17 b(mixtures.)24 b(The)17 b(second)g(section)g(establishes)h(a)e(connection)h(b)q(et)o(w)o(een)g (EM)f(and)h(the)g(gradien)o(t)f(of)0 677 y(the)11 b(log)h(lik)o(eliho)q (o)q(d.)21 b(W)l(e)12 b(then)f(presen)o(t)h(a)e(comparativ)o(e)h (discussion)i(of)e(the)h(adv)m(an)o(tages)e(and)i(disadv)m(an)o(tages)f (of)0 746 y(v)m(arious)i(optimization)h(algorithms)f(in)g(the)g (Gaussian)g(mixture)g(setting.)19 b(W)l(e)13 b(then)g(presen)o(t)g (empirical)h(results)0 815 y(suggesting)i(that)g(EM)g(regularizes)h (the)g(condition)g(n)o(um)o(b)q(er)g(of)f(the)g(e\013ectiv)o(e)h (Hessian.)24 b(The)16 b(fourth)g(section)0 884 y(presen)o(ts)f(a)f (theoretical)h(analysis)h(of)e(this)h(empirical)i(\014nding.)k(The)15 b(\014nal)g(section)g(presen)o(ts)g(our)f(conclusions.)0 1040 y Fq(2)69 b(The)23 b(EM)g(algorithm)f(for)h(Gaussian)h(mixtures)0 1154 y Fw(W)l(e)15 b(study)h(the)f(follo)o(wing)h(probabilistic)h(mo)q (del:)684 1270 y Fo(P)6 b Fw(\()p Fo(x)p Fm(j)p Fw(\002\))12 b(=)904 1218 y Fn(K)890 1230 y Fe(X)889 1321 y Fn(j)r Fp(=1)958 1270 y Fo(\013)987 1277 y Fn(j)1006 1270 y Fo(P)6 b Fw(\()p Fo(x)p Fm(j)p Fo(m)1138 1277 y Fn(j)1156 1270 y Fo(;)i Fw(\006)1210 1277 y Fn(j)1227 1270 y Fw(\))p Fo(;)633 b Fw(\(1\))0 1391 y(and)518 1460 y Fo(P)6 b Fw(\()p Fo(x)p Fm(j)p Fo(m)650 1467 y Fn(j)668 1460 y Fo(;)i Fw(\006)722 1467 y Fn(j)739 1460 y Fw(\))13 b(=)907 1429 y(1)p 823 1450 192 2 v 823 1465 a Fm(p)p 860 1465 51 2 v 860 1502 a Fw(2)p Fo(\031)r Fm(j)p Fw(\006)957 1509 y Fn(j)974 1502 y Fm(j)992 1471 y Fj(1)p 992 1477 16 2 v 992 1497 a(2)1019 1460 y Fo(e)1040 1439 y Fl(\000)1072 1425 y Fj(1)p 1073 1431 V 1073 1452 a(2)1093 1439 y Fp(\()p Fn(x)p Fl(\000)p Fn(m)1185 1444 y Ff(j)1202 1439 y Fp(\))1216 1427 y Ff(T)1240 1439 y Fp(\006)1265 1424 y Fk(\000)p Fj(1)1265 1450 y Ff(j)1306 1439 y Fp(\()p Fn(x)p Fl(\000)p Fn(m)1398 1444 y Ff(j)1415 1439 y Fp(\))0 1577 y Fw(where)h Fo(\013)159 1584 y Fn(j)190 1577 y Fm(\025)f Fw(0)g(and)360 1545 y Fe(P)404 1558 y Fn(K)404 1589 y(j)r Fp(=1)475 1577 y Fo(\013)504 1584 y Fn(j)535 1577 y Fw(=)g(1.)19 b(The)14 b(parameter)e(v)o(ector)h(\002)h(consists)g(of)f(the)g(mixing) i(prop)q(ortions)e Fo(\013)1841 1584 y Fn(j)1859 1577 y Fw(,)h(the)0 1646 y(mean)h(v)o(ectors)g Fo(m)316 1653 y Fn(j)334 1646 y Fw(,)g(and)g(the)g(co)o(v)m(ariance)h(matrices)f (\006)963 1653 y Fn(j)981 1646 y Fw(.)71 1715 y(Giv)o(en)22 b Fo(K)j Fw(and)d(giv)o(en)h Fo(N)k Fw(indep)q(enden)o(t,)f(iden)o (tically)e(distributed)g(samples)f Fm(f)p Fo(x)1529 1699 y Fp(\()p Fn(t)p Fp(\))1571 1715 y Fm(g)1594 1699 y Fn(N)1594 1727 y Fp(1)1627 1715 y Fw(,)g(w)o(e)f(obtain)h(the)0 1784 y(follo)o(wing)16 b(log)f(lik)o(eliho)q(o)q(d:)467 1767 y Fp(2)541 1900 y Fo(l)q Fw(\(\002\))d(=)h(log)766 1847 y Fn(N)756 1860 y Fe(Y)753 1951 y Fn(t)p Fp(=1)819 1900 y Fo(P)6 b Fw(\()p Fo(x)898 1882 y Fp(\()p Fn(t)p Fp(\))940 1900 y Fm(j)p Fw(\002\))12 b(=)1081 1847 y Fn(N)1066 1860 y Fe(X)1067 1951 y Fn(t)p Fp(=1)1134 1900 y Fw(log)c Fo(P)e Fw(\()p Fo(x)1279 1882 y Fp(\()p Fn(t)p Fp(\))1322 1900 y Fm(j)p Fw(\002\))p Fo(;)490 b Fw(\(2\))0 2017 y(whic)o(h)17 b(can)g(b)q(e)g(optimized)h(via)f(the)f(follo)o (wing)i(iterativ)o(e)e(algorithm)h(\(see,)f(e.g,)g(Dempster,)g(Laird)i (&)e(Rubin,)0 2086 y(1977\):)484 2210 y Fo(\013)513 2186 y Fp(\()p Fn(k)q Fp(+1\))513 2223 y Fn(j)648 2210 y Fw(=)730 2142 y Fe(P)774 2155 y Fn(N)774 2185 y(t)p Fp(=1)841 2174 y Fo(h)867 2150 y Fp(\()p Fn(k)q Fp(\))867 2187 y Fn(j)916 2174 y Fw(\()p Fo(t)p Fw(\))p 730 2200 239 2 v 828 2241 a Fo(N)473 2344 y(m)513 2320 y Fp(\()p Fn(k)q Fp(+1\))513 2356 y Fn(j)648 2344 y Fw(=)730 2275 y Fe(P)774 2288 y Fn(N)774 2318 y(t)p Fp(=1)841 2307 y Fo(h)867 2283 y Fp(\()p Fn(k)q Fp(\))867 2320 y Fn(j)916 2307 y Fw(\()p Fo(t)p Fw(\))p Fo(x)994 2290 y Fp(\()p Fn(t)p Fp(\))p 730 2333 307 2 v 764 2355 a Fe(P)808 2368 y Fn(N)808 2398 y(t)p Fp(=1)875 2387 y Fo(h)901 2363 y Fp(\()p Fn(k)q Fp(\))901 2400 y Fn(j)950 2387 y Fw(\()p Fo(t)p Fw(\))480 2508 y(\006)513 2484 y Fp(\()p Fn(k)q Fp(+1\))513 2521 y Fn(j)648 2508 y Fw(=)730 2440 y Fe(P)774 2453 y Fn(N)774 2483 y(t)p Fp(=1)841 2472 y Fo(h)867 2448 y Fp(\()p Fn(k)q Fp(\))867 2485 y Fn(j)916 2472 y Fw(\()p Fo(t)p Fw(\)[)p Fo(x)1007 2455 y Fp(\()p Fn(t)p Fp(\))1059 2472 y Fm(\000)10 b Fo(m)1144 2448 y Fp(\()p Fn(k)q Fp(\))1144 2485 y Fn(j)1193 2472 y Fw(][)p Fo(x)1245 2455 y Fp(\()p Fn(t)p Fp(\))1297 2472 y Fm(\000)g Fo(m)1382 2448 y Fp(\()p Fn(k)q Fp(\))1382 2485 y Fn(j)1431 2472 y Fw(])1444 2455 y Fn(T)p 730 2498 742 2 v 947 2519 a Fe(P)991 2533 y Fn(N)991 2563 y(t)p Fp(=1)1059 2552 y Fo(h)1085 2528 y Fp(\()p Fn(k)q Fp(\))1085 2565 y Fn(j)1134 2552 y Fw(\()p Fo(t)p Fw(\))p Fo(x)1212 2538 y Fp(\()p Fn(t)p Fp(\))1891 2508 y Fw(\(3\))p 0 2594 780 2 v 52 2621 a Fj(2)69 2637 y Fi(Although)19 b(w)o(e)d(fo)q(cus)h(on)h(maxim)o(um)g(lik)o(eliho)q(o)r(d)i(\(ML\))d (estimation)i(in)e(this)h(pap)q(er,)h(it)e(is)h(straigh)o(tforw)o(ard)g (to)f(apply)h(our)0 2693 y(results)c(to)f(maxim)o(um)h(a)f(p)q (osteriori)i(\(MAP\))e(estimation)i(b)o(y)e(m)o(ultiplyi)q(ng)j(the)d (lik)o(elih)q(o)q(o)q(d)j(b)o(y)e(a)f(prior.)963 2817 y Fw(3)p eop %%Page: 4 4 4 3 bop 0 202 a Fw(where)15 b(the)h(p)q(osterior)f(probabilities)i Fo(h)687 178 y Fp(\()p Fn(k)q Fp(\))687 215 y Fn(j)751 202 y Fw(are)e(de\014ned)i(as)e(follo)o(ws:)604 357 y Fo(h)630 333 y Fp(\()p Fn(k)q Fp(\))630 370 y Fn(j)679 357 y Fw(\()p Fo(t)p Fw(\))d(=)852 321 y Fo(\013)881 297 y Fp(\()p Fn(k)q Fp(\))881 333 y Fn(j)930 321 y Fo(P)6 b Fw(\()p Fo(x)1009 304 y Fp(\()p Fn(t)p Fp(\))1051 321 y Fm(j)p Fo(m)1104 297 y Fp(\()p Fn(k)q Fp(\))1104 333 y Fn(j)1152 321 y Fo(;)i Fw(\006)1206 297 y Fp(\()p Fn(k)q Fp(\))1206 333 y Fn(j)1254 321 y Fw(\))p 796 347 531 2 v 796 368 a Fe(P)840 381 y Fn(K)840 412 y(i)p Fp(=1)907 400 y Fo(\013)936 376 y Fp(\()p Fn(k)q Fp(\))936 413 y Fn(i)985 400 y Fo(P)e Fw(\()p Fo(x)1064 387 y Fp(\()p Fn(t)p Fp(\))1106 400 y Fm(j)p Fo(m)1159 376 y Fp(\()p Fn(k)q Fp(\))1159 413 y Fn(i)1208 400 y Fo(;)i Fw(\006)1262 376 y Fp(\()p Fn(k)q Fp(\))1262 413 y Fn(i)1309 400 y Fw(\))1332 357 y Fo(:)0 528 y Fq(3)69 b(Connection)22 b(b)r(et)n(w)n(een)f(EM)i(and)h(gradien)n(t)f(ascen)n(t)0 641 y Fw(In)e(the)g(follo)o(wing)h(theorem)e(w)o(e)h(establish)h(a)e (relationship)i(b)q(et)o(w)o(een)f(the)g(gradien)o(t)g(of)f(the)h(log)g (lik)o(eliho)q(o)q(d)0 710 y(and)16 b(the)h(step)f(in)h(parameter)f (space)g(tak)o(en)g(b)o(y)g(the)h(EM)f(algorithm.)23 b(In)17 b(particular)g(w)o(e)f(sho)o(w)f(that)h(the)g(EM)0 779 y(step)g(can)g(b)q(e)g(obtained)h(b)o(y)f(prem)o(ultiplying)i(the)e (gradien)o(t)g(b)o(y)f(a)h(p)q(ositiv)o(e)h(de\014nite)g(matrix.)22 b(W)l(e)15 b(pro)o(vide)i(an)0 848 y(explicit)g(expression)f(for)f(the) g(matrix.)0 967 y Fd(Theorem)i(1)23 b Fx(A)o(t)16 b(e)n(ach)g(iter)n (ation)g(of)h(the)f(EM)g(algorithm)h(Eq.)j(\(3\),)c(we)g(have)658 1090 y Fm(A)694 1071 y Fp(\()p Fn(k)q Fp(+1\))799 1090 y Fm(\000)10 b(A)880 1071 y Fp(\()p Fn(k)q Fp(\))971 1090 y Fw(=)42 b Fo(P)1083 1066 y Fp(\()p Fn(k)q Fp(\))1077 1104 y Fl(A)1148 1059 y Fo(@)s(l)p 1137 1080 63 2 v 1137 1121 a(@)s Fm(A)1205 1090 y(j)1218 1103 y Fl(A)p Fp(=)p Fl(A)1301 1093 y Fj(\()p Ff(k)q Fj(\))1891 1090 y Fw(\(4\))651 1200 y Fo(m)691 1176 y Fp(\()p Fn(k)q Fp(+1\))691 1213 y Fn(j)795 1200 y Fm(\000)10 b Fo(m)880 1176 y Fp(\()p Fn(k)q Fp(\))880 1213 y Fn(j)971 1200 y Fw(=)42 b Fo(P)1083 1182 y Fp(\()p Fn(k)q Fp(\))1077 1212 y Fn(m)1108 1217 y Ff(j)1159 1170 y Fo(@)s(l)p 1137 1190 85 2 v 1137 1231 a(@)s(m)1204 1238 y Fn(j)1227 1200 y Fm(j)1240 1221 y Fn(m)1271 1226 y Ff(j)1287 1221 y Fp(=)p Fn(m)1345 1203 y Fj(\()p Ff(k)q Fj(\))1345 1233 y Ff(j)1891 1200 y Fw(\(5\))489 1324 y(v)o(ec[\006)598 1300 y Fp(\()p Fn(k)q Fp(+1\))598 1337 y Fn(j)691 1324 y Fw(])10 b Fm(\000)g Fw(v)o(ec[\006)868 1300 y Fp(\()p Fn(k)q Fp(\))868 1337 y Fn(j)917 1324 y Fw(])41 b(=)h Fo(P)1083 1300 y Fp(\()p Fn(k)q Fp(\))1077 1337 y(\006)1102 1342 y Ff(j)1199 1293 y Fo(@)s(l)p 1137 1313 167 2 v 1137 1355 a(@)s Fw(v)o(ec[\006)1273 1362 y Fn(j)1290 1355 y Fw(])1308 1324 y Fm(j)1321 1344 y Fp(\006)1346 1349 y Ff(j)1362 1344 y Fp(=\006)1414 1327 y Fj(\()p Ff(k)q Fj(\))1414 1356 y Ff(j)1891 1324 y Fw(\(6\))0 1450 y Fx(wher)n(e)497 1564 y Fo(P)532 1540 y Fp(\()p Fn(k)q Fp(\))526 1578 y Fl(A)623 1564 y Fw(=)714 1533 y(1)p 705 1554 42 2 v 705 1595 a Fo(N)751 1564 y Fm(f)p Fw(diag)q([)p Fo(\013)900 1540 y Fp(\()p Fn(k)q Fp(\))900 1576 y(1)949 1564 y Fo(;)8 b Fm(\001)g(\001)g(\001)t Fo(;)g(\013)1079 1540 y Fp(\()p Fn(k)q Fp(\))1079 1578 y Fn(K)1128 1564 y Fw(])h Fm(\000)i(A)1232 1545 y Fp(\()p Fn(k)q Fp(\))1281 1564 y Fw(\()p Fm(A)1335 1545 y Fp(\()p Fn(k)q Fp(\))1384 1564 y Fw(\))1402 1545 y Fn(T)1429 1564 y Fm(g)439 b Fw(\(7\))497 1697 y Fo(P)532 1679 y Fp(\()p Fn(k)q Fp(\))526 1709 y Fn(m)557 1714 y Ff(j)623 1697 y Fw(=)783 1661 y(\006)816 1637 y Fp(\()p Fn(k)q Fp(\))816 1674 y Fn(j)p 705 1687 239 2 v 705 1708 a Fe(P)749 1722 y Fn(N)749 1752 y(t)p Fp(=1)816 1741 y Fo(h)842 1717 y Fp(\()p Fn(k)q Fp(\))842 1754 y Fn(j)891 1741 y Fw(\()p Fo(t)p Fw(\))1891 1697 y(\(8\))497 1837 y Fo(P)532 1813 y Fp(\()p Fn(k)q Fp(\))526 1850 y(\006)551 1855 y Ff(j)623 1837 y Fw(=)812 1806 y(2)p 705 1826 V 705 1848 a Fe(P)749 1861 y Fn(N)749 1891 y(t)p Fp(=1)816 1880 y Fo(h)842 1856 y Fp(\()p Fn(k)q Fp(\))842 1893 y Fn(j)891 1880 y Fw(\()p Fo(t)p Fw(\))948 1837 y(\006)981 1813 y Fp(\()p Fn(k)q Fp(\))981 1850 y Fn(j)1040 1837 y Fm(\012)10 b Fw(\006)1118 1813 y Fp(\()p Fn(k)q Fp(\))1118 1850 y Fn(j)1891 1837 y Fw(\(9\))0 1988 y Fx(wher)n(e)15 b Fm(A)h Fx(denotes)f(the)h(ve)n(ctor)f(of)h(mixing)e(pr)n(op)n (ortions)i Fw([)p Fo(\013)1031 1995 y Fp(1)1050 1988 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(\013)1181 1995 y Fn(K)1215 1988 y Fw(])1228 1972 y Fn(T)1255 1988 y Fx(,)15 b Fo(j)j Fx(indexes)d(the)g(mixtur)n(e)h(c)n(omp)n(onents)0 2057 y(\()p Fo(j)i Fw(=)f(1)p Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(K)s Fx(\),)18 b Fo(k)h Fx(denotes)f(the)h(iter)n(ation)f(numb)n (er,)h(\\)p Fw(v)o(ec)p Fx([B]")g(denotes)f(the)h(ve)n(ctor)f(obtaine)n (d)g(by)h(stacking)0 2126 y(the)e(c)n(olumn)f(ve)n(ctors)g(of)h(the)g (matrix)h Fo(B)r Fx(,)f(and)g(\\)p Fm(\012)p Fx(")h(denotes)e(the)h(Kr) n(one)n(cker)f(pr)n(o)n(duct.)22 b(Mor)n(e)n(over,)17 b(given)f(the)0 2195 y(c)n(onstr)n(aints)233 2163 y Fe(P)277 2176 y Fn(K)277 2206 y(j)r Fp(=1)348 2195 y Fo(\013)377 2171 y Fp(\()p Fn(k)q Fp(\))377 2208 y Fn(j)444 2195 y Fw(=)h(1)i Fx(and)g Fo(\013)658 2171 y Fp(\()p Fn(k)q Fp(\))658 2208 y Fn(j)725 2195 y Fm(\025)e Fw(0)p Fx(,)j Fo(P)869 2171 y Fp(\()p Fn(k)q Fp(\))863 2209 y Fl(A)937 2195 y Fx(is)e(a)h(p)n(ositive)g(de\014nite)f(matrix)i(and)f(the)g (matric)n(es)g Fo(P)1900 2171 y Fp(\()p Fn(k)q Fp(\))1894 2200 y Fn(m)1925 2205 y Ff(j)0 2264 y Fx(and)d Fo(P)123 2240 y Fp(\()p Fn(k)q Fp(\))117 2278 y(\006)142 2283 y Ff(j)189 2264 y Fx(ar)n(e)g(p)n(ositive)g(de\014nite)f(with)i(pr)n (ob)n(ability)f(one)g(for)h Fo(N)j Fx(su\016ciently)c(lar)n(ge.)71 2442 y Fd(Pro)q(of.)j(\(1\))c Fw(W)l(e)g(b)q(egin)h(b)o(y)e (considering)i(the)f(EM)f(up)q(date)h(for)f(the)h(mixing)g(prop)q (ortions)g Fo(\013)1698 2449 y Fn(i)1712 2442 y Fw(.)20 b(F)l(rom)13 b(Eqs.)0 2510 y(\(1\))h(and)i(\(2\),)d(w)o(e)i(ha)o(v)o(e) 500 2621 y Fo(@)s(l)p 489 2641 63 2 v 489 2683 a(@)s Fm(A)557 2651 y(j)570 2664 y Fl(A)p Fp(=)p Fl(A)653 2654 y Fj(\()p Ff(k)q Fj(\))711 2651 y Fw(=)774 2598 y Fn(N)759 2611 y Fe(X)760 2702 y Fn(t)p Fp(=1)832 2621 y Fw([)p Fo(P)6 b Fw(\()p Fo(x)924 2604 y Fp(\()p Fn(t)p Fp(\))966 2621 y Fo(;)i(\022)1009 2597 y Fp(\()p Fn(k)q Fp(\))1008 2633 y(1)1057 2621 y Fw(\))p Fo(;)g Fm(\001)g(\001)g(\001)d Fo(;)j(P)e Fw(\()p Fo(x)1256 2604 y Fp(\()p Fn(t)p Fp(\))1298 2621 y Fo(;)i(\022)1341 2597 y Fp(\()p Fn(k)q Fp(\))1340 2634 y Fn(K)1389 2621 y Fw(\)])1420 2604 y Fn(T)p 832 2641 616 2 v 930 2663 a Fe(P)974 2676 y Fn(K)974 2706 y(i)p Fp(=1)1040 2695 y Fo(\013)1069 2671 y Fp(\()p Fn(k)q Fp(\))1069 2708 y Fn(i)1118 2695 y Fo(P)e Fw(\()p Fo(x)1197 2681 y Fp(\()p Fn(t)p Fp(\))1240 2695 y Fo(;)i(\022)1283 2671 y Fp(\()p Fn(k)q Fp(\))1282 2708 y Fn(i)1331 2695 y Fw(\))1452 2651 y Fo(:)963 2817 y Fw(4)p eop %%Page: 5 5 5 4 bop 0 202 a Fw(Prem)o(ultiplying)17 b(b)o(y)e Fo(P)409 178 y Fp(\()p Fn(k)q Fp(\))403 216 y Fl(A)459 202 y Fw(,)g(w)o(e)f (obtain)6 338 y Fo(P)41 314 y Fp(\()p Fn(k)q Fp(\))35 351 y Fl(A)106 307 y Fo(@)s(l)p 95 327 63 2 v 95 369 a(@)s Fm(A)163 338 y(j)176 350 y Fl(A)p Fp(=)p Fl(A)259 341 y Fj(\()p Ff(k)q Fj(\))346 338 y Fw(=)437 307 y(1)p 428 327 42 2 v 428 369 a Fo(N)496 285 y Fn(N)482 297 y Fe(X)483 388 y Fn(t)p Fp(=1)554 307 y Fm(f)p Fw([)p Fo(\013)619 283 y Fp(\()p Fn(k)q Fp(\))619 319 y(1)668 307 y Fo(P)6 b Fw(\()p Fo(x)747 290 y Fp(\()p Fn(t)p Fp(\))789 307 y Fo(;)i(\022)832 283 y Fp(\()p Fn(k)q Fp(\))831 319 y(1)881 307 y Fw(\))p Fo(;)g Fm(\001)g(\001)g(\001)t Fo(;)g(\013)1029 283 y Fp(\()p Fn(k)q Fp(\))1029 320 y Fn(K)1077 307 y Fo(P)e Fw(\()p Fo(x)1156 290 y Fp(\()p Fn(t)p Fp(\))1199 307 y Fo(;)i(\022)1242 283 y Fp(\()p Fn(k)q Fp(\))1241 320 y Fn(K)1290 307 y Fw(\)])1321 290 y Fn(T)1358 307 y Fm(\000)j(A)1440 290 y Fp(\()p Fn(k)q Fp(\))1496 275 y Fe(P)1540 288 y Fn(K)1540 318 y(i)p Fp(=1)1607 307 y Fo(\013)1636 283 y Fp(\()p Fn(k)q Fp(\))1636 320 y Fn(i)1685 307 y Fo(P)6 b Fw(\()p Fo(x)1764 290 y Fp(\()p Fn(t)p Fp(\))1806 307 y Fo(;)i(\022)1849 283 y Fp(\()p Fn(k)q Fp(\))1848 320 y Fn(i)1898 307 y Fw(\))p Fm(g)p 554 327 1385 2 v 1037 349 a Fe(P)1080 362 y Fn(K)1080 392 y(i)p Fp(=1)1147 381 y Fo(\013)1176 357 y Fp(\()p Fn(k)q Fp(\))1176 394 y Fn(i)1225 381 y Fo(P)e Fw(\()p Fo(x)1304 368 y Fp(\()p Fn(t)p Fp(\))1347 381 y Fo(;)i(\022)1390 357 y Fp(\()p Fn(k)q Fp(\))1389 394 y Fn(i)1438 381 y Fw(\))346 490 y(=)437 459 y(1)p 428 480 42 2 v 428 521 a Fo(N)496 437 y Fn(N)482 450 y Fe(X)483 540 y Fn(t)p Fp(=1)542 490 y Fw([)p Fo(h)581 466 y Fp(\()p Fn(k)q Fp(\))581 503 y(1)629 490 y Fw(\()p Fo(t)p Fw(\))p Fo(;)g Fm(\001)g(\001)g(\001)d Fo(;)j(h)809 466 y Fp(\()p Fn(k)q Fp(\))809 504 y Fn(K)857 490 y Fw(\()p Fo(t)p Fw(\)])922 471 y Fn(T)959 490 y Fm(\000)i(A)1040 471 y Fp(\()p Fn(k)q Fp(\))1090 490 y Fo(:)0 620 y Fw(The)15 b(up)q(date)h(form)o(ula)f(for) g Fm(A)g Fw(in)h(Eq.)k(\(3\))14 b(can)h(b)q(e)h(rewritten)f(as)469 753 y Fm(A)505 735 y Fp(\()p Fn(k)q Fp(+1\))612 753 y Fw(=)e Fm(A)696 735 y Fp(\()p Fn(k)q Fp(\))755 753 y Fw(+)815 723 y(1)p 805 743 V 805 785 a Fo(N)874 700 y Fn(N)859 713 y Fe(X)860 804 y Fn(t)p Fp(=1)919 753 y Fw([)p Fo(h)958 729 y Fp(\()p Fn(k)q Fp(\))958 766 y(1)1007 753 y Fw(\()p Fo(t)p Fw(\))p Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(h)1187 729 y Fp(\()p Fn(k)q Fp(\))1187 767 y Fn(K)1235 753 y Fw(\()p Fo(t)p Fw(\)])1300 735 y Fn(T)1337 753 y Fm(\000)i(A)1418 735 y Fp(\()p Fn(k)q Fp(\))1467 753 y Fo(:)0 884 y Fw(Com)o(bining)17 b(the)g(last)f(t)o(w)o(o)f(equations) i(establishes)g(the)g(up)q(date)g(rule)g(for)f Fm(A)h Fw(\(Eq.)23 b(4\).)f(F)l(urthermore,)16 b(for)g(an)0 952 y(arbitrary)10 b(v)o(ector)h Fo(u)p Fw(,)g(w)o(e)g(ha)o(v)o(e)g Fo(N)5 b(u)604 936 y Fn(T)631 952 y Fo(P)666 928 y Fp(\()p Fn(k)q Fp(\))660 966 y Fl(A)715 952 y Fo(u)13 b Fw(=)g Fo(u)828 936 y Fn(T)855 952 y Fw(diag)q([)p Fo(\013)981 928 y Fp(\()p Fn(k)q Fp(\))981 965 y(1)1030 952 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(\013)1161 928 y Fp(\()p Fn(k)q Fp(\))1161 966 y Fn(K)1209 952 y Fw(])p Fo(u)r Fm(\000)r Fw(\()p Fo(u)1331 936 y Fn(T)1358 952 y Fm(A)1394 936 y Fp(\()p Fn(k)q Fp(\))1443 952 y Fw(\))1461 936 y Fp(2)1481 952 y Fw(.)18 b(By)11 b(Jensen's)h(inequalit)o(y)0 1021 y(w)o(e)j(ha)o(v)o(e)569 1143 y Fo(u)595 1124 y Fn(T)623 1143 y Fw(diag)q([)p Fo(\013)749 1119 y Fp(\()p Fn(k)q Fp(\))749 1155 y(1)797 1143 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(\013)928 1119 y Fp(\()p Fn(k)q Fp(\))928 1157 y Fn(K)976 1143 y Fw(])p Fo(u)42 b Fw(=)1148 1090 y Fn(K)1134 1102 y Fe(X)1133 1194 y Fn(j)r Fp(=1)1202 1143 y Fo(\013)1231 1119 y Fp(\()p Fn(k)q Fp(\))1231 1156 y Fn(j)1280 1143 y Fo(u)1306 1124 y Fp(2)1306 1154 y Fn(j)1057 1301 y Fo(>)f Fw(\()1166 1248 y Fn(K)1152 1260 y Fe(X)1151 1351 y Fn(j)r Fp(=1)1220 1301 y Fo(\013)1249 1277 y Fp(\()p Fn(k)q Fp(\))1249 1313 y Fn(j)1298 1301 y Fo(u)1324 1308 y Fn(j)1342 1301 y Fw(\))1360 1282 y Fp(2)1057 1422 y Fw(=)g(\()p Fo(u)1177 1403 y Fn(T)1205 1422 y Fm(A)1241 1403 y Fp(\()p Fn(k)q Fp(\))1290 1422 y Fw(\))1308 1403 y Fp(2)1327 1422 y Fo(:)0 1529 y Fw(Th)o(us,)16 b Fo(u)155 1512 y Fn(T)183 1529 y Fo(P)218 1505 y Fp(\()p Fn(k)q Fp(\))212 1542 y Fl(A)267 1529 y Fo(u)f(>)g Fw(0)h(and)g Fo(P)521 1505 y Fp(\()p Fn(k)q Fp(\))515 1542 y Fl(A)587 1529 y Fw(is)h(p)q(ositiv)o(e)g(de\014nite)h(giv)o(en)f(the)f (constrain)o(ts)1398 1497 y Fe(P)1442 1510 y Fn(K)1442 1540 y(j)r Fp(=1)1513 1529 y Fo(\013)1542 1505 y Fp(\()p Fn(k)q Fp(\))1542 1542 y Fn(j)1605 1529 y Fw(=)f(1)h(and)h Fo(\013)1813 1505 y Fp(\()p Fn(k)q Fp(\))1813 1542 y Fn(j)1876 1529 y Fm(\025)e Fw(0)0 1598 y(for)g(all)h Fo(j)s Fw(.)71 1666 y Fd(\(2\))g Fw(W)l(e)f(no)o(w)f(consider)j(the)e (EM)g(up)q(date)g(for)g(the)g(means)g Fo(m)1148 1673 y Fn(i)1162 1666 y Fw(.)20 b(It)c(follo)o(ws)f(from)f(Eqs.)20 b(\(1\))14 b(and)i(\(2\))e(that)537 1769 y Fo(@)s(l)p 515 1789 85 2 v 515 1830 a(@)s(m)582 1837 y Fn(j)605 1799 y Fm(j)618 1820 y Fn(m)649 1825 y Ff(j)665 1820 y Fp(=)p Fn(m)723 1802 y Fj(\()p Ff(k)q Fj(\))723 1832 y Ff(j)782 1799 y Fw(=)844 1746 y Fn(N)830 1759 y Fe(X)831 1849 y Fn(t)p Fp(=1)897 1799 y Fo(h)923 1775 y Fp(\()p Fn(k)q Fp(\))923 1812 y Fn(j)972 1799 y Fw(\()p Fo(t)p Fw(\)\(\006)1075 1775 y Fp(\()p Fn(k)q Fp(\))1075 1812 y Fn(j)1123 1799 y Fw(\))1141 1781 y Fl(\000)p Fp(1)1188 1799 y Fw([)p Fo(x)1227 1781 y Fp(\()p Fn(t)p Fp(\))1279 1799 y Fm(\000)c Fo(m)1364 1775 y Fp(\()p Fn(k)q Fp(\))1364 1812 y Fn(j)1413 1799 y Fw(])p Fo(:)0 1944 y Fw(Prem)o(ultiplying)17 b(b)o(y)e Fo(P)409 1920 y Fp(\()p Fn(k)q Fp(\))403 1950 y Fn(m)434 1955 y Ff(j)474 1944 y Fw(yields)413 2081 y Fo(P)448 2062 y Fp(\()p Fn(k)q Fp(\))442 2092 y Fn(m)473 2097 y Ff(j)524 2050 y Fo(@)s(l)p 502 2070 V 502 2112 a(@)s(m)569 2119 y Fn(j)592 2081 y Fm(j)605 2102 y Fn(m)636 2107 y Ff(j)652 2102 y Fp(=)p Fn(m)710 2084 y Fj(\()p Ff(k)q Fj(\))710 2113 y Ff(j)797 2081 y Fw(=)987 2050 y(1)p 879 2070 239 2 v 879 2092 a Fe(P)923 2105 y Fn(N)923 2136 y(t)p Fp(=1)990 2124 y Fo(h)1016 2100 y Fp(\()p Fn(k)q Fp(\))1016 2137 y Fn(j)1065 2124 y Fw(\()p Fo(t)p Fw(\))1144 2028 y Fn(N)1129 2041 y Fe(X)1131 2131 y Fn(t)p Fp(=1)1197 2081 y Fo(h)1223 2057 y Fp(\()p Fn(k)q Fp(\))1223 2094 y Fn(j)1272 2081 y Fw(\()p Fo(t)p Fw(\))p Fo(x)1350 2062 y Fp(\()p Fn(t)p Fp(\))1402 2081 y Fm(\000)c Fo(m)1488 2057 y Fp(\()p Fn(k)q Fp(\))1488 2094 y Fn(j)797 2209 y Fw(=)42 b Fo(m)914 2185 y Fp(\()p Fn(k)q Fp(+1\))914 2222 y Fn(j)1018 2209 y Fm(\000)10 b Fo(m)1103 2185 y Fp(\()p Fn(k)q Fp(\))1103 2222 y Fn(j)1152 2209 y Fo(:)0 2316 y Fw(F)l(rom)17 b(Eq.)28 b(\(3\),)17 b(w)o(e)g(ha)o(v)o(e)483 2284 y Fe(P)527 2297 y Fn(N)527 2327 y(t)p Fp(=1)595 2316 y Fo(h)621 2292 y Fp(\()p Fn(k)q Fp(\))621 2329 y Fn(j)670 2316 y Fw(\()p Fo(t)p Fw(\))f Fo(>)i Fw(0;)g(moreo)o(v)o (er,)f(\006)1090 2292 y Fp(\()p Fn(k)q Fp(\))1090 2329 y Fn(j)1156 2316 y Fw(is)i(p)q(ositiv)o(e)f(de\014nite)h(with)g (probabilit)o(y)g(one)0 2385 y(assuming)14 b(that)g Fo(N)k Fw(is)d(large)f(enough)g(suc)o(h)h(that)e(the)h(matrix)g(is)g(of)g (full)i(rank.)j(Th)o(us,)14 b(it)g(follo)o(ws)g(from)f(Eq.)20 b(\(8\))0 2453 y(that)14 b Fo(P)133 2430 y Fp(\()p Fn(k)q Fp(\))127 2459 y Fn(m)158 2464 y Ff(j)198 2453 y Fw(is)i(p)q(ositiv)o (e)g(de\014nite)g(with)g(probabilit)o(y)g(one.)71 2522 y Fd(\(3\))g Fw(Finally)l(,)g(w)o(e)f(pro)o(v)o(e)g(the)g(third)h(part) e(of)h(the)g(theorem.)20 b(It)15 b(follo)o(ws)g(from)g(Eqs.)k(\(1\))c (and)g(\(2\))f(that)190 2624 y Fo(@)s(l)p 172 2645 78 2 v 172 2686 a(@)s Fw(\006)232 2693 y Fn(j)255 2655 y Fm(j)268 2676 y Fp(\006)293 2681 y Ff(j)309 2676 y Fp(=\006)361 2658 y Fj(\()p Ff(k)q Fj(\))361 2688 y Ff(j)420 2655 y Fw(=)f Fm(\000)508 2624 y Fw(1)p 508 2645 23 2 v 508 2686 a(2)558 2602 y Fn(N)543 2615 y Fe(X)544 2705 y Fn(t)p Fp(=1)611 2655 y Fo(h)637 2631 y Fp(\()p Fn(k)q Fp(\))637 2668 y Fn(j)686 2655 y Fw(\()p Fo(t)p Fw(\)\(\006)789 2631 y Fp(\()p Fn(k)q Fp(\))789 2668 y Fn(j)837 2655 y Fw(\))855 2636 y Fl(\000)p Fp(1)902 2655 y Fm(f)p Fw(\006)958 2631 y Fp(\()p Fn(k)q Fp(\))958 2668 y Fn(j)1016 2655 y Fm(\000)e Fw([)p Fo(x)1101 2636 y Fp(\()p Fn(t)p Fp(\))1153 2655 y Fm(\000)f Fo(m)1238 2631 y Fp(\()p Fn(k)q Fp(\))1238 2668 y Fn(j)1287 2655 y Fw(][)p Fo(x)1339 2636 y Fp(\()p Fn(t)p Fp(\))1390 2655 y Fm(\000)h Fo(m)1476 2631 y Fp(\()p Fn(k)q Fp(\))1476 2668 y Fn(j)1524 2655 y Fw(\)])1555 2636 y Fn(T)1582 2655 y Fm(g)p Fw(\(\006)1656 2631 y Fp(\()p Fn(k)q Fp(\))1656 2668 y Fn(j)1704 2655 y Fw(\))1722 2636 y Fl(\000)p Fp(1)1769 2655 y Fo(:)963 2817 y Fw(5)p eop %%Page: 6 6 6 5 bop 0 202 a Fw(With)15 b(this)h(in)g(mind,)g(w)o(e)f(rewrite)g(the) g(EM)g(up)q(date)h(form)o(ula)f(for)f(\006)1197 178 y Fp(\()p Fn(k)q Fp(\))1197 215 y Fn(j)1261 202 y Fw(as)238 349 y(\006)271 325 y Fp(\()p Fn(k)q Fp(+1\))271 362 y Fn(j)407 349 y Fw(=)41 b(\006)516 325 y Fp(\()p Fn(k)q Fp(\))516 362 y Fn(j)575 349 y Fw(+)733 319 y(1)p 626 339 239 2 v 626 361 a Fe(P)669 374 y Fn(N)669 404 y(t)p Fp(=1)737 393 y Fo(h)763 369 y Fp(\()p Fn(k)q Fp(\))763 406 y Fn(j)812 393 y Fw(\()p Fo(t)p Fw(\))890 296 y Fn(N)876 309 y Fe(X)877 400 y Fn(t)p Fp(=1)944 349 y Fo(h)970 325 y Fp(\()p Fn(k)q Fp(\))970 362 y Fn(j)1019 349 y Fw(\()p Fo(t)p Fw(\)[)p Fo(x)1110 331 y Fp(\()p Fn(t)p Fp(\))1161 349 y Fm(\000)11 b Fo(m)1247 325 y Fp(\()p Fn(k)q Fp(\))1247 362 y Fn(j)1295 349 y Fw(][)p Fo(x)1347 331 y Fp(\()p Fn(t)p Fp(\))1399 349 y Fm(\000)g Fo(m)1485 325 y Fp(\()p Fn(k)q Fp(\))1485 362 y Fn(j)1533 349 y Fw(])1546 331 y Fn(T)1583 349 y Fm(\000)g Fw(\006)1662 325 y Fp(\()p Fn(k)q Fp(\))1662 362 y Fn(j)407 514 y Fw(=)41 b(\006)516 490 y Fp(\()p Fn(k)q Fp(\))516 527 y Fn(j)575 514 y Fw(+)692 478 y(2\006)748 454 y Fp(\()p Fn(k)q Fp(\))748 490 y Fn(j)p 626 504 V 626 525 a Fe(P)669 538 y Fn(N)669 569 y(t)p Fp(=1)737 557 y Fo(h)763 533 y Fp(\()p Fn(k)q Fp(\))763 570 y Fn(j)812 557 y Fw(\()p Fo(t)p Fw(\))869 514 y Fo(V)896 521 y Fp(\006)921 526 y Ff(j)939 514 y Fw(\006)972 490 y Fp(\()p Fn(k)q Fp(\))972 527 y Fn(j)1020 514 y Fo(;)0 656 y Fw(where)238 784 y Fo(V)265 791 y Fp(\006)290 796 y Ff(j)350 784 y Fw(=)h Fm(\000)467 753 y Fw(1)p 467 773 23 2 v 467 815 a(2)517 731 y Fn(N)503 743 y Fe(X)504 834 y Fn(t)p Fp(=1)570 784 y Fo(h)596 760 y Fp(\()p Fn(k)q Fp(\))596 797 y Fn(j)645 784 y Fw(\()p Fo(t)p Fw(\)\(\006)748 760 y Fp(\()p Fn(k)q Fp(\))748 797 y Fn(j)796 784 y Fw(\))814 765 y Fl(\000)p Fp(1)861 784 y Fm(f)p Fw(\006)917 760 y Fp(\()p Fn(k)q Fp(\))917 797 y Fn(j)975 784 y Fm(\000)11 b Fw([)p Fo(x)1060 765 y Fp(\()p Fn(t)p Fp(\))1112 784 y Fm(\000)f Fo(m)1197 760 y Fp(\()p Fn(k)q Fp(\))1197 797 y Fn(j)1246 784 y Fw(][)p Fo(x)1298 765 y Fp(\()p Fn(t)p Fp(\))1349 784 y Fm(\000)h Fo(m)1435 760 y Fp(\()p Fn(k)q Fp(\))1435 797 y Fn(j)1484 784 y Fw(])1497 765 y Fn(T)1524 784 y Fm(g)p Fw(\(\006)1598 760 y Fp(\()p Fn(k)q Fp(\))1598 797 y Fn(j)1646 784 y Fw(\))1664 765 y Fl(\000)p Fp(1)350 917 y Fw(=)450 886 y Fo(@)s(l)p 432 906 78 2 v 432 948 a(@)s Fw(\006)492 955 y Fn(j)515 917 y Fm(j)528 938 y Fp(\006)553 943 y Ff(j)569 938 y Fp(=\006)621 920 y Fj(\()p Ff(k)q Fj(\))621 949 y Ff(j)667 917 y Fo(:)0 1041 y Fw(That)k(is,)g(w)o (e)g(ha)o(v)o(e)521 1139 y(\006)554 1115 y Fp(\()p Fn(k)q Fp(+1\))554 1152 y Fn(j)660 1139 y Fw(=)e(\006)741 1115 y Fp(\()p Fn(k)q Fp(\))741 1152 y Fn(j)800 1139 y Fw(+)917 1103 y(2\006)973 1079 y Fp(\()p Fn(k)q Fp(\))973 1115 y Fn(j)p 850 1129 239 2 v 850 1150 a Fe(P)894 1163 y Fn(N)894 1194 y(t)p Fp(=1)962 1182 y Fo(h)988 1158 y Fp(\()p Fn(k)q Fp(\))988 1195 y Fn(j)1037 1182 y Fw(\()p Fo(t)p Fw(\))1117 1108 y Fo(@)s(l)p 1098 1129 78 2 v 1098 1170 a(@)s Fw(\006)1158 1177 y Fn(j)1181 1139 y Fm(j)1194 1160 y Fp(\006)1219 1165 y Ff(j)1236 1160 y Fp(=\006)1288 1142 y Fj(\()p Ff(k)q Fj(\))1288 1172 y Ff(j)1334 1139 y Fw(\006)1367 1115 y Fp(\()p Fn(k)q Fp(\))1367 1152 y Fn(j)1415 1139 y Fo(:)0 1271 y Fw(Utilizing)18 b(the)d(iden)o(tit)o(y)h(v)o(ec[)p Fo(AB)r(C)s Fw(])c(=)h(\()p Fo(C)741 1255 y Fn(T)778 1271 y Fm(\012)e Fo(A)p Fw(\)v)o(ec[)p Fo(B)r Fw(],)k(w)o(e)f(obtain)346 1393 y(v)o(ec)q([\006)456 1369 y Fp(\()p Fn(k)q Fp(+1\))456 1406 y Fn(j)549 1393 y Fw(])e(=)h(v)o(ec[\006)731 1369 y Fp(\()p Fn(k)q Fp(\))731 1406 y Fn(j)779 1393 y Fw(])d(+)960 1362 y(2)p 853 1382 239 2 v 853 1404 a Fe(P)896 1417 y Fn(N)896 1447 y(t)p Fp(=1)964 1436 y Fo(h)990 1412 y Fp(\()p Fn(k)q Fp(\))990 1449 y Fn(j)1039 1436 y Fw(\()p Fo(t)p Fw(\))1096 1393 y(\(\006)1147 1369 y Fp(\()p Fn(k)q Fp(\))1147 1406 y Fn(j)1205 1393 y Fm(\012)g Fw(\006)1283 1369 y Fp(\()p Fn(k)q Fp(\))1283 1406 y Fn(j)1332 1393 y Fw(\))1373 1362 y Fo(@)s(l)p 1355 1382 78 2 v 1355 1424 a(@)s Fw(\006)1415 1431 y Fn(j)1437 1393 y Fm(j)1450 1413 y Fp(\006)1475 1418 y Ff(j)1492 1413 y Fp(=\006)1544 1396 y Fj(\()p Ff(k)q Fj(\))1544 1425 y Ff(j)1590 1393 y Fo(:)0 1552 y Fw(Th)o(us)15 b Fo(P)150 1528 y Fp(\()p Fn(k)q Fp(\))144 1566 y(\006)169 1571 y Ff(j)212 1552 y Fw(=)359 1534 y Fp(2)p 265 1541 206 2 v 265 1550 a Fe(P)309 1561 y Ff(N)309 1594 y(t)p Fj(=1)367 1579 y Fn(h)387 1562 y Fj(\()p Ff(k)q Fj(\))387 1591 y Ff(j)431 1579 y Fp(\()p Fn(t)p Fp(\))476 1552 y Fw(\(\006)527 1528 y Fp(\()p Fn(k)q Fp(\))527 1565 y Fn(j)585 1552 y Fm(\012)c Fw(\006)664 1528 y Fp(\()p Fn(k)q Fp(\))664 1565 y Fn(j)712 1552 y Fw(\).)20 b(Moreo)o(v)o(er,)13 b(for)i(an)g(arbitrary)f(matrix)h Fo(U)5 b Fw(,)15 b(w)o(e)g(ha)o(v)o(e)382 1693 y(v)o(ec[)p Fo(U)5 b Fw(])507 1674 y Fn(T)534 1693 y Fw(\(\006)585 1669 y Fp(\()p Fn(k)q Fp(\))585 1706 y Fn(j)644 1693 y Fm(\012)10 b Fw(\006)722 1669 y Fp(\()p Fn(k)q Fp(\))722 1706 y Fn(j)771 1693 y Fw(\)v)o(ec)o([)p Fo(U)5 b Fw(])41 b(=)h(tr\(\006)1118 1669 y Fp(\()p Fn(k)q Fp(\))1118 1706 y Fn(j)1166 1693 y Fo(U)5 b Fw(\006)1235 1669 y Fp(\()p Fn(k)q Fp(\))1235 1706 y Fn(j)1283 1693 y Fo(U)1319 1674 y Fn(T)1347 1693 y Fw(\))954 1774 y(=)42 b(tr\(\(\006)1136 1750 y Fp(\()p Fn(k)q Fp(\))1136 1787 y Fn(j)1184 1774 y Fo(U)5 b Fw(\))1238 1755 y Fn(T)1265 1774 y Fw(\(\006)1316 1750 y Fp(\()p Fn(k)q Fp(\))1316 1787 y Fn(j)1364 1774 y Fo(U)g Fw(\)\))954 1855 y(=)42 b(v)o(ec[\006)1140 1831 y Fp(\()p Fn(k)q Fp(\))1140 1868 y Fn(j)1188 1855 y Fo(U)5 b Fw(])1237 1836 y Fn(T)1265 1855 y Fw(v)o(ec[\006)1374 1831 y Fp(\()p Fn(k)q Fp(\))1374 1868 y Fn(j)1422 1855 y Fo(U)g Fw(])12 b Fm(\025)h Fw(0)p Fo(;)0 1968 y Fw(where)18 b(equalit)o(y)g(holds)g(only)g(when)g(\006)686 1944 y Fp(\()p Fn(k)q Fp(\))686 1981 y Fn(j)735 1968 y Fo(U)j Fw(=)c(0)g(for)g(all)i Fo(U)5 b Fw(.)27 b(Equalit)o(y)18 b(is)g(imp)q(ossible,)i(ho)o(w)o(ev)o(er,)d(since)h(\006)1900 1944 y Fp(\()p Fn(k)q Fp(\))1900 1981 y Fn(j)0 2037 y Fw(is)g(p)q(ositiv)o(e)g(de\014nite)h(with)f(probabilit)o(y)g(one)g Fo(N)k Fw(is)c(su\016cien)o(tly)g(large.)27 b(Th)o(us)17 b(it)h(follo)o(ws)f(from)g(Eq.)26 b(\(9\))17 b(and)0 2073 y Fe(P)44 2087 y Fn(N)44 2117 y(t)p Fp(=1)111 2106 y Fo(h)137 2082 y Fp(\()p Fn(k)q Fp(\))137 2118 y Fn(j)186 2106 y Fw(\()p Fo(t)p Fw(\))c Fo(>)g Fw(0)i(that)f Fo(P)470 2082 y Fp(\()p Fn(k)q Fp(\))464 2119 y(\006)489 2124 y Ff(j)534 2106 y Fw(is)i(p)q(ositiv)o(e)g(de\014nite)h(with)e (probabilit)o(y)i(one.)588 b Fc(2)71 2243 y Fw(Using)13 b(the)g(notation)g(\002)g(=)g([)p Fo(m)601 2227 y Fn(T)601 2255 y Fp(1)628 2243 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(m)770 2227 y Fn(T)770 2256 y(K)803 2243 y Fo(;)g Fw(v)o(ec)o([\006)932 2250 y Fp(1)952 2243 y Fw(])965 2227 y Fn(T)992 2243 y Fo(;)g Fm(\001)g(\001)g(\001)d Fo(;)j Fw(v)o(ec)o([\006)1202 2250 y Fn(K)1236 2243 y Fw(])1249 2227 y Fn(T)1276 2243 y Fo(;)g Fm(A)1333 2227 y Fn(T)1360 2243 y Fw(])1373 2227 y Fn(T)1400 2243 y Fw(,)13 b(and)g Fo(P)6 b Fw(\(\002\))13 b(=)g(diag)q([)p Fo(P)1805 2250 y Fn(m)1836 2255 y Fj(1)1855 2243 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)0 2312 y(P)29 2319 y Fn(m)60 2325 y Ff(K)92 2312 y Fo(;)j(P)142 2319 y Fp(\006)167 2324 y Fj(1)186 2312 y Fw(,)15 b Fm(\001)8 b(\001)g(\001)d Fo(;)j(P)324 2319 y Fp(\006)349 2325 y Ff(K)381 2312 y Fo(;)g(P)431 2319 y Fl(A)460 2312 y Fw(],)15 b(w)o(e)g(can)g(com)o (bine)h(the)f(three)h(up)q(dates)f(in)h(Theorem)f(1)g(in)o(to)g(a)g (single)i(equation:)616 2435 y(\002)651 2416 y Fp(\()p Fn(k)q Fp(+1\))758 2435 y Fw(=)c(\002)841 2416 y Fp(\()p Fn(k)q Fp(\))900 2435 y Fw(+)e Fo(P)6 b Fw(\(\002)1034 2416 y Fp(\()p Fn(k)q Fp(\))1083 2435 y Fw(\))1116 2404 y Fo(@)s(l)p 1106 2424 63 2 v 1106 2466 a(@)s Fw(\002)1173 2435 y Fm(j)1186 2448 y Fp(\002=\002)1267 2438 y Fj(\()p Ff(k)q Fj(\))1313 2435 y Fo(;)542 b Fw(\(10\))0 2555 y(Under)21 b(the)f(conditions)i(of)e(Theorem)g(1,)h Fo(P)6 b Fw(\(\002)848 2539 y Fp(\()p Fn(k)q Fp(\))897 2555 y Fw(\))20 b(is)h(a)f(p)q(ositiv)o(e)h(de\014nite)h(matrix)e(with)g (probabilit)o(y)i(one.)0 2624 y(Recalling)i(that)d(for)g(a)g(p)q (ositiv)o(e)h(de\014nite)h(matrix)f Fo(B)r Fw(,)h(w)o(e)e(ha)o(v)o(e) 1196 2606 y Fn(@)r(l)p 1188 2613 48 2 v 1188 2640 a(@)r Fp(\002)1241 2596 y Fn(T)1268 2624 y Fo(B)1318 2606 y Fn(@)r(l)p 1310 2613 V 1310 2640 a(@)r Fp(\002)1386 2624 y Fo(>)j Fw(0,)e(w)o(e)f(ha)o(v)o(e)g(the)h(follo)o(wing)0 2693 y(corollary:)963 2817 y(6)p eop %%Page: 7 7 7 6 bop 0 195 a Fd(Corollary)18 b(1)k Fx(F)m(or)14 b(e)n(ach)g(iter)n (ation)g(of)g(the)g(EM)g(algorithm)g(given)g(by)g(Eq.\(3\),)f(the)h(se) n(ar)n(ch)g(dir)n(e)n(ction)f Fw(\002)1814 179 y Fp(\()p Fn(k)q Fp(+1\))1913 195 y Fm(\000)0 264 y Fw(\002)35 247 y Fp(\()p Fn(k)q Fp(\))100 264 y Fx(has)k(a)f(p)n(ositive)g(pr)n (oje)n(ction)g(on)g(the)g(gr)n(adient)g(of)h Fo(l)q Fx(.)71 378 y Fw(That)h(is,)i(the)f(EM)f(algorithm)h(can)g(b)q(e)g(view)o(ed)h (as)e(a)h(v)m(ariable)h(metric)f(gradien)o(t)g(ascen)o(t)g(algorithm)f (for)0 447 y(whic)o(h)12 b(the)f(pro)s(jection)f(matrix)h Fo(P)6 b Fw(\(\002)643 431 y Fp(\()p Fn(k)q Fp(\))692 447 y Fw(\))k(c)o(hanges)h(at)f(eac)o(h)h(iteration)g(as)g(a)f (function)i(of)e(the)h(curren)o(t)g(parameter)0 516 y(v)m(alue)16 b(\002)152 500 y Fp(\()p Fn(k)q Fp(\))202 516 y Fw(.)71 585 y(Our)c(results)h(extend)g(earlier)g(results)g(due)g(to)f(Baum)g (and)h(Sell)h(\(1968\).)j(Baum)12 b(and)h(Sell)h(studied)f(recursiv)o (e)0 654 y(equations)j(of)e(the)i(follo)o(wing)g(form:)123 788 y Fo(x)149 770 y Fp(\()p Fn(k)q Fp(+1\))256 788 y Fw(=)d Fo(T)6 b Fw(\()p Fo(x)381 770 y Fp(\()p Fn(k)q Fp(\))429 788 y Fw(\))p Fo(;)37 b(T)6 b Fw(\()p Fo(x)574 770 y Fp(\()p Fn(k)q Fp(\))622 788 y Fw(\))13 b(=)g([)p Fo(T)6 b Fw(\()p Fo(x)791 770 y Fp(\()p Fn(k)q Fp(\))839 788 y Fw(\))857 795 y Fp(1)876 788 y Fo(;)i Fm(\001)g(\001)g(\001)d Fo(;)j(T)e Fw(\()p Fo(x)1055 770 y Fp(\()p Fn(k)q Fp(\))1102 788 y Fw(\))1120 795 y Fn(K)1154 788 y Fw(])p Fo(;)37 b(T)6 b Fw(\()p Fo(x)1294 770 y Fp(\()p Fn(k)q Fp(\))1343 788 y Fw(\)\))1379 795 y Fn(i)1405 788 y Fw(=)1513 758 y Fo(x)1539 734 y Fp(\()p Fn(k)q Fp(\))1539 771 y Fn(i)1588 758 y Fo(@)s(J)r(=@)s(x)1718 734 y Fp(\()p Fn(k)q Fp(\))1718 771 y Fn(i)p 1458 778 364 2 v 1458 800 a Fe(P)1501 813 y Fn(K)1501 843 y(i)p Fp(=1)1568 832 y Fo(x)1594 808 y Fp(\()p Fn(k)q Fp(\))1594 845 y Fn(i)1643 832 y Fo(@)s(J)r(=@)s(x) 1773 808 y Fp(\()p Fn(k)q Fp(\))1773 845 y Fn(i)0 940 y Fw(where)17 b Fo(x)159 916 y Fp(\()p Fn(k)q Fp(\))159 952 y Fn(i)222 940 y Fm(\025)e Fw(0)p Fo(;)331 907 y Fe(P)375 921 y Fn(K)375 951 y(i)p Fp(=1)442 940 y Fo(x)468 916 y Fp(\()p Fn(k)q Fp(\))468 952 y Fn(i)531 940 y Fw(=)g(1,)h(where)g Fo(J)21 b Fw(is)c(a)f(p)q(olynomial)i(in)f Fo(x)1213 916 y Fp(\()p Fn(k)q Fp(\))1213 952 y Fn(i)1278 940 y Fw(ha)o(ving)g(p)q(ositiv)o(e)g(co)q(e\016cien)o(ts.)25 b(They)0 1008 y(sho)o(w)o(ed)16 b(that)g(the)h(searc)o(h)f(direction)i (of)e(this)i(recursiv)o(e)f(form)o(ula,)f(i.e.,)h Fo(T)6 b Fw(\()p Fo(x)1349 992 y Fp(\()p Fn(k)q Fp(\))1397 1008 y Fw(\))11 b Fm(\000)g Fo(x)1498 992 y Fp(\()p Fn(k)q Fp(\))1547 1008 y Fw(,)16 b(has)h(a)f(p)q(ositiv)o(e)i(pro-)0 1077 y(jection)j(on)g(the)f(gradien)o(t)h(of)f(of)g Fo(J)25 b Fw(with)c(resp)q(ect)g(to)f(the)g Fo(x)1093 1061 y Fp(\()p Fn(k)q Fp(\))1162 1077 y Fw(\(see)h(also)f(Levinson,)j(Rabiner) f(&)f(Sondhi,)0 1146 y(1983\).)29 b(It)18 b(can)h(b)q(e)g(sho)o(wn)f (that)g(Baum)h(and)f(Sell's)i(recursiv)o(e)f(form)o(ula)g(implies)h (the)f(EM)f(up)q(date)h(form)o(ula)0 1215 y(for)f Fm(A)g Fw(in)h(a)f(Gaussian)g(mixture.)30 b(Th)o(us,)18 b(the)h(\014rst)f (statemen)o(t)f(in)i(Theorem)f(1)g(is)h(a)f(sp)q(ecial)i(case)e(of)g (Baum)0 1284 y(and)f(Sell's)h(earlier)g(w)o(ork.)24 b(Ho)o(w)o(ev)o (er,)16 b(Baum)h(and)g(Sell's)h(theorem)f(is)g(an)g(existence)h (theorem)f(and)g(do)q(es)g(not)0 1353 y(pro)o(vide)i(an)g(explicit)h (expression)g(for)e(the)h(matrix)f Fo(P)953 1360 y Fl(A)1002 1353 y Fw(that)g(transforms)f(the)i(gradien)o(t)f(direction)i(in)o(to)f (the)0 1422 y(EM)c(direction.)21 b(Our)16 b(theorem)f(pro)o(vides)h (suc)o(h)f(an)g(explicit)j(form)c(for)h Fo(P)1278 1429 y Fl(A)1308 1422 y Fw(.)21 b(Moreo)o(v)o(er,)13 b(w)o(e)i(generalize)i (Baum)0 1491 y(and)e(Sell's)h(results)f(to)f(handle)i(the)f(up)q(dates) g(for)f Fo(m)906 1498 y Fn(j)939 1491 y Fw(and)g(\006)1059 1498 y Fn(j)1078 1491 y Fw(,)g(and)h(w)o(e)f(pro)o(vide)h(explicit)i (expressions)f(for)e(the)0 1559 y(p)q(ositiv)o(e)i(de\014nite)h (transformation)d(matrices)h Fo(P)851 1566 y Fn(m)882 1571 y Ff(j)916 1559 y Fw(and)g Fo(P)1033 1566 y Fp(\006)1058 1571 y Ff(j)1092 1559 y Fw(as)g(w)o(ell.)71 1628 y(It)f(is)h(also)f(w)o (orth)o(while)h(to)e(compare)h(the)h(EM)f(algorithm)g(to)g(other)g (gradien)o(t-based)g(optimization)i(meth-)0 1697 y(o)q(ds.)22 b Fx(Newton)-5 b('s)17 b(metho)n(d)k Fw(is)c(obtained)f(b)o(y)g(prem)o (ultiplying)j(the)d(gradien)o(t)g(b)o(y)f(the)i(in)o(v)o(erse)f(of)f (the)i(Hessian)f(of)0 1766 y(the)f(log)h(lik)o(eliho)q(o)q(d:)639 1841 y(\002)674 1823 y Fp(\()p Fn(k)q Fp(+1\))781 1841 y Fw(=)d(\002)864 1823 y Fp(\()p Fn(k)q Fp(\))923 1841 y Fw(+)d Fo(H)t Fw(\(\002)1063 1823 y Fp(\()p Fn(k)q Fp(\))1112 1841 y Fw(\))1130 1823 y Fl(\000)p Fp(1)1216 1811 y Fo(@)s(l)p 1182 1831 111 2 v 1182 1874 a(@)s Fw(\002)1244 1861 y Fp(\()p Fn(k)q Fp(\))1297 1841 y Fo(:)558 b Fw(\(11\))0 1935 y(Newton's)21 b(metho)q(d)i(is)g(the)f(metho)q(d)g(of)g(c)o(hoice) h(when)g(it)f(can)g(b)q(e)h(applied,)j(but)c(the)g(algorithm)g(is)h (often)0 2004 y(di\016cult)e(to)e(use)h(in)h(practice.)33 b(In)21 b(particular,)g(the)e(algorithm)h(can)g(div)o(erge)g(when)g (the)g(Hessian)g(b)q(ecomes)0 2073 y(nearly)d(singular;)h(moreo)o(v)o (er,)d(the)h(computational)h(costs)f(of)g(computing)h(the)f(in)o(v)o (erse)h(Hessian)g(at)f(eac)o(h)h(step)0 2142 y(can)e(b)q(e)g (considerable.)22 b(An)15 b(alternativ)o(e)g(is)g(to)g(appro)o(ximate)f (the)h(in)o(v)o(erse)g(b)o(y)g(a)f(recursiv)o(ely)i(up)q(dated)g (matrix)0 2211 y Fo(B)36 2194 y Fp(\()p Fn(k)q Fp(+1\))143 2211 y Fw(=)d Fo(B)227 2194 y Fp(\()p Fn(k)q Fp(\))281 2211 y Fw(+)5 b Fo(\021)r Fw(\001)p Fo(B)420 2194 y Fp(\()p Fn(k)q Fp(\))469 2211 y Fw(.)19 b(Suc)o(h)13 b(a)f(mo)q(di\014cation)i (is)f(called)h(a)e Fx(quasi-Newton)i(metho)n(d)p Fw(.)20 b(Con)o(v)o(en)o(tional)12 b(quasi-)0 2280 y(Newton)h(metho)q(ds)g(are) g(unconstrained)h(optimization)g(metho)q(ds,)f(ho)o(w)o(ev)o(er,)f(and) i(m)o(ust)e(b)q(e)i(mo)q(di\014ed)g(in)g(order)0 2349 y(to)g(b)q(e)h(used)g(in)g(the)g(mixture)g(setting)f(\(where)g(there)h (are)f(probabilistic)j(constrain)o(ts)d(on)g(the)h(parameters\).)j(In)0 2417 y(addition,)f(quasi-Newton)g(metho)q(ds)g(generally)g(require)h (that)d(a)i(one-dimensional)h(searc)o(h)f(b)q(e)g(p)q(erformed)f(at)0 2486 y(eac)o(h)i(iteration)g(in)h(order)f(to)f(guaran)o(tee)h(con)o(v)o (ergence.)28 b(The)18 b(EM)g(algorithm)g(can)g(b)q(e)h(view)o(ed)g(as)e (a)h(sp)q(ecial)0 2555 y(form)c(of)f(quasi-Newton)i(metho)q(d)g(in)g (whic)o(h)g(the)f(pro)s(jection)h(matrix)f Fo(P)6 b Fw(\(\002)1316 2539 y Fp(\()p Fn(k)q Fp(\))1365 2555 y Fw(\))14 b(in)h(Eq.)20 b(\(10\))13 b(pla)o(ys)h(the)h(role)f(of)0 2624 y Fo(B)36 2608 y Fp(\()p Fn(k)q Fp(\))86 2624 y Fw(.)19 b(As)13 b(w)o(e)f(discuss)j(in)f(the)f(remainder)g(of)g(the)g(pap)q(er,)h(this) f(particular)h(matrix)f(has)g(a)f(n)o(um)o(b)q(er)i(of)e(fa)o(v)o (orable)0 2693 y(prop)q(erties)k(that)e(mak)o(e)h(EM)g(particularly)h (attractiv)o(e)e(for)h(optimization)h(in)g(the)g(mixture)f(setting.)963 2817 y(7)p eop %%Page: 8 8 8 7 bop 0 195 a Fq(4)69 b(Constrained)23 b(optimizati)o(on)e(and)j (general)e(con)n(v)n(ergence)0 309 y Fw(An)f(imp)q(ortan)o(t)e(prop)q (ert)o(y)h(of)g(the)g(matrix)g Fo(P)27 b Fw(is)21 b(that)e(the)h(EM)g (step)g(in)h(parameter)f(space)g(automatically)0 378 y(satis\014es)d(the)f(probabilistic)j(constrain)o(ts)d(of)g(the)g (mixture)h(mo)q(del)g(in)h(Eq.)23 b(\(1\).)g(The)16 b(domain)h(of)f (\002)h(con)o(tains)0 447 y(t)o(w)o(o)e(regions)i(that)f(em)o(b)q(o)q (dy)i(the)e(probabilistic)j(constrain)o(ts:)k Fm(D)1146 454 y Fp(1)1181 447 y Fw(=)15 b Fm(f)p Fw(\002)g(:)1332 415 y Fe(P)1376 428 y Fn(K)1376 458 y(j)r Fp(=1)1447 447 y Fo(\013)1476 423 y Fp(\()p Fn(k)q Fp(\))1476 460 y Fn(j)1540 447 y Fw(=)h(1)p Fm(g)g Fw(and)h Fm(D)1778 454 y Fp(2)1812 447 y Fw(=)f Fm(f)p Fw(\002)f(:)0 516 y Fo(\013)29 492 y Fp(\()p Fn(k)q Fp(\))29 529 y Fn(j)95 516 y Fm(\025)i Fw(0,)h(\006)234 523 y Fn(j)270 516 y Fw(p)q(ositiv)o(e)e(de\014nite)q Fm(g)p Fw(.)28 b(F)l(or)17 b(the)h(EM)f(algorithm)h(the)g(up)q(date)g(for)f(the)h(mixing)h(prop)q (ortions)e Fo(\013)1930 523 y Fn(j)0 584 y Fw(can)e(b)q(e)h(rewritten)f (as)g(follo)o(ws:)543 709 y Fo(\013)572 685 y Fp(\()p Fn(k)q Fp(+1\))572 722 y Fn(j)678 709 y Fw(=)741 679 y(1)p 731 699 42 2 v 731 741 a Fo(N)800 657 y Fn(N)785 669 y Fe(X)786 760 y Fn(t)p Fp(=1)913 673 y Fo(\013)942 649 y Fp(\()p Fn(k)q Fp(\))942 686 y Fn(j)991 673 y Fo(P)6 b Fw(\()p Fo(x)1070 656 y Fp(\()p Fn(t)p Fp(\))1113 673 y Fm(j)p Fo(m)1166 649 y Fp(\()p Fn(k)q Fp(\))1166 686 y Fn(j)1214 673 y Fo(;)i Fw(\006)1268 649 y Fp(\()p Fn(k)q Fp(\))1268 686 y Fn(j)1316 673 y Fw(\))p 858 699 531 2 v 858 721 a Fe(P)902 734 y Fn(K)902 764 y(i)p Fp(=1)968 753 y Fo(\013)997 729 y Fp(\()p Fn(k)q Fp(\))997 766 y Fn(i)1046 753 y Fo(P)e Fw(\()p Fo(x)1125 740 y Fp(\()p Fn(t)p Fp(\))1168 753 y Fm(j)p Fo(m)1221 729 y Fp(\()p Fn(k)q Fp(\))1221 766 y Fn(i)1269 753 y Fo(;)i Fw(\006)1323 729 y Fp(\()p Fn(k)q Fp(\))1323 766 y Fn(i)1371 753 y Fw(\))1394 709 y Fo(:)0 837 y Fw(It)15 b(is)h(ob)o(vious)f(that)g(the)g (iteration)h(sta)o(ys)e(within)i Fm(D)911 844 y Fp(1)931 837 y Fw(.)k(Similarly)l(,)d(the)e(up)q(date)h(for)f(\006)1500 844 y Fn(j)1533 837 y Fw(can)g(b)q(e)h(rewritten)f(as:)197 975 y(\006)230 951 y Fp(\()p Fn(k)q Fp(+1\))230 988 y Fn(j)337 975 y Fw(=)497 944 y(1)p 390 964 239 2 v 390 986 a Fe(P)433 999 y Fn(N)433 1030 y(t)p Fp(=1)501 1018 y Fo(h)527 994 y Fp(\()p Fn(k)q Fp(\))527 1031 y Fn(j)576 1018 y Fw(\()p Fo(t)p Fw(\))654 922 y Fn(N)640 935 y Fe(X)641 1025 y Fn(t)p Fp(=1)768 938 y Fo(\013)797 915 y Fp(\()p Fn(k)q Fp(\))797 951 y Fn(j)846 938 y Fo(P)6 b Fw(\()p Fo(x)925 922 y Fp(\()p Fn(t)p Fp(\))967 938 y Fm(j)p Fo(m)1020 915 y Fp(\()p Fn(k)q Fp(\))1020 951 y Fn(j)1069 938 y Fo(;)i Fw(\006)1123 915 y Fp(\()p Fn(k)q Fp(\))1123 951 y Fn(j)1171 938 y Fw(\))p 713 964 531 2 v 713 986 a Fe(P)757 999 y Fn(K)757 1030 y(i)p Fp(=1)823 1018 y Fo(\013)852 994 y Fp(\()p Fn(k)q Fp(\))852 1031 y Fn(i)901 1018 y Fo(P)e Fw(\()p Fo(x)980 1005 y Fp(\()p Fn(t)p Fp(\))1023 1018 y Fm(j)p Fo(m)1076 994 y Fp(\()p Fn(k)q Fp(\))1076 1031 y Fn(i)1124 1018 y Fo(;)i Fw(\006)1178 994 y Fp(\()p Fn(k)q Fp(\))1178 1031 y Fn(i)1226 1018 y Fw(\))1248 975 y([)p Fo(x)1287 956 y Fp(\()p Fn(t)p Fp(\))1339 975 y Fm(\000)j Fo(m)1425 951 y Fp(\()p Fn(k)q Fp(\))1425 988 y Fn(j)1474 975 y Fw(][)p Fo(x)1526 956 y Fp(\()p Fn(t)p Fp(\))1577 975 y Fm(\000)g Fo(m)1663 951 y Fp(\()p Fn(k)q Fp(\))1663 988 y Fn(j)1711 975 y Fw(])1724 956 y Fn(T)0 1109 y Fw(whic)o(h)16 b(sta)o(ys)e(within)j Fm(D)421 1116 y Fp(2)456 1109 y Fw(for)d Fo(N)20 b Fw(su\016cien)o(tly) c(large.)71 1178 y(Whereas)21 b(EM)f(automatically)i(satis\014es)g(the) f(probabilistic)j(constrain)o(ts)c(of)h(a)g(mixture)h(mo)q(del,)h (other)0 1247 y(optimization)17 b(tec)o(hniques)g(generally)f(require)h (mo)q(di\014cation)g(to)e(satisfy)g(the)h(constrain)o(ts.)21 b(One)16 b(approac)o(h)f(is)0 1315 y(to)j(mo)q(dify)g(eac)o(h)h (iterativ)o(e)f(step)g(to)g(k)o(eep)h(the)f(parameters)f(within)j(the)e (constrained)h(domain.)29 b(A)18 b(n)o(um)o(b)q(er)0 1384 y(of)d(suc)o(h)g(tec)o(hniques)h(ha)o(v)o(e)f(b)q(een)h(dev)o (elop)q(ed,)g(including)i(feasible)e(direction)h(metho)q(ds,)d(activ)o (e)h(sets,)g(gradien)o(t)0 1453 y(pro)s(jection,)i(reduced-gradien)o (t,)i(and)f(linearly-constrained)h(quasi-Newton.)27 b(These)18 b(constrained)g(metho)q(ds)0 1522 y(all)24 b(incur)f(extra)f (computational)h(costs)f(to)g(c)o(hec)o(k)h(and)g(main)o(tain)g(the)f (constrain)o(ts)h(and,)h(moreo)o(v)o(er,)f(the)0 1591 y(theoretical)e(con)o(v)o(ergence)f(rates)g(for)g(suc)o(h)g (constrained)h(algorithms)f(need)i(not)e(b)q(e)h(the)f(same)g(as)g (that)f(for)0 1660 y(the)f(corresp)q(onding)h(unconstrained)g (algorithms.)27 b(A)18 b(second)h(approac)o(h)e(is)i(to)e(transform)f (the)i(constrained)0 1729 y(optimization)c(problem)h(in)o(to)e(an)h (unconstrained)g(problem)g(b)q(efore)g(using)g(the)g(unconstrained)h (metho)q(d.)k(This)0 1798 y(can)d(b)q(e)g(accomplished)i(via)e(p)q (enalt)o(y)h(and)f(barrier)g(functions,)g(Lagrangian)g(terms,)f(or)g (re-parameterization.)0 1866 y(Once)j(again,)e(the)h(extra)f (algorithmic)h(mac)o(hinery)g(renders)h(simple)g(comparisons)e(based)h (on)g(unconstrained)0 1935 y(con)o(v)o(ergence)f(rates)f(problematic.) 23 b(Moreo)o(v)o(er,)14 b(it)i(is)h(not)e(easy)h(to)f(meet)h(the)g (constrain)o(ts)f(on)h(the)g(co)o(v)m(ariance)0 2004 y(matrices)f(in)h(the)g(mixture)f(using)h(suc)o(h)g(tec)o(hniques.)71 2073 y(A)22 b(second)h(app)q(ealing)h(prop)q(ert)o(y)f(of)f Fo(P)6 b Fw(\(\002)835 2057 y Fp(\()p Fn(k)q Fp(\))884 2073 y Fw(\))22 b(is)h(that)f(eac)o(h)h(iteration)g(of)f(EM)g(is)h (guaran)o(teed)f(to)g(in-)0 2142 y(crease)d(the)f(lik)o(eliho)q(o)q(d)k (\(i.e.,)d Fo(l)q Fw(\(\002)606 2125 y Fp(\()p Fn(k)q Fp(+1\))699 2142 y Fw(\))f Fm(\025)h Fo(l)q Fw(\(\002)857 2125 y Fp(\()p Fn(k)q Fp(\))905 2142 y Fw(\)\).)29 b(This)19 b(monotonic)g(con)o(v)o(ergence)g(of)f(the)h(lik)o(eliho)q(o)q(d)i(is)0 2211 y(ac)o(hiev)o(ed)f(without)f(step-size)h(parameters)e(or)g(line)j (searc)o(hes.)31 b(Other)19 b(gradien)o(t-based)h(optimization)g(tec)o (h-)0 2280 y(niques,)h(including)h(gradien)o(t)d(descen)o(t,)i (quasi-Newton,)f(and)g(Newton's)e(metho)q(d,)i(do)g(not)f(pro)o(vide)g (suc)o(h)h(a)0 2349 y(simple)g(theoretical)f(guaran)o(tee,)f(ev)o(en)h (assuming)g(that)e(the)i(constrained)g(problem)g(has)f(b)q(een)i (transformed)0 2417 y(in)o(to)g(an)g(unconstrained)h(one.)35 b(F)l(or)19 b(gradien)o(t)h(ascen)o(t,)h(the)f(step)g(size)h Fo(\021)g Fw(m)o(ust)e(b)q(e)i(c)o(hosen)f(to)g(ensure)g(that)0 2486 y Fm(k)p Fw(\002)58 2470 y Fp(\()p Fn(k)q Fp(+1\))162 2486 y Fm(\000)10 b Fw(\002)242 2470 y Fp(\()p Fn(k)q Fl(\000)p Fp(1\))337 2486 y Fm(k)p Fo(=)p Fm(k)p Fw(\(\002)459 2470 y Fp(\()p Fn(k)q Fp(\))517 2486 y Fm(\000)g Fw(\002)597 2470 y Fp(\()p Fn(k)q Fl(\000)p Fp(1\))691 2486 y Fw(\))p Fm(k)17 b(\024)g(k)p Fo(I)e Fw(+)e Fo(\021)r(H)t Fw(\(\002)1027 2470 y Fp(\()p Fn(k)q Fl(\000)p Fp(1\))1119 2486 y Fw(\)\))p Fm(k)j Fo(<)i Fw(1.)27 b(This)18 b(requires)h(a)e(one-dimensional)0 2555 y(line)h(searc)o(h)e(or)f(an)h(optimization)h(of)f Fo(\021)h Fw(at)e(eac)o(h)h(iteration,)h(whic)o(h)g(requires)f(extra)g (computation)g(whic)o(h)h(can)0 2624 y(slo)o(w)e(do)o(wn)g(the)g(con)o (v)o(ergence.)20 b(An)c(alternativ)o(e)f(is)h(to)f(\014x)g Fo(\021)h Fw(to)f(a)g(v)o(ery)g(small)h(v)m(alue)g(whic)o(h)g (generally)g(mak)o(es)0 2693 y Fm(k)p Fo(I)8 b Fw(+)d Fo(\021)r(H)t Fw(\(\002)211 2676 y Fp(\()p Fn(k)q Fl(\000)p Fp(1\))304 2693 y Fw(\)\))p Fm(k)12 b Fw(close)h(to)f(one)g(and)h (results)g(in)h(slo)o(w)e(con)o(v)o(ergence.)19 b(F)l(or)12 b(Newton's)g(metho)q(d,)h(the)g(iterativ)o(e)963 2817 y(8)p eop %%Page: 9 9 9 8 bop 0 195 a Fw(pro)q(cess)16 b(is)h(usually)g(required)g(to)e(b)q (e)i(near)f(a)f(solution,)i(otherwise)f(the)g(Hessian)h(ma)o(y)e(b)q(e) h(inde\014nite)j(and)d(the)0 264 y(iteration)i(ma)o(y)f(not)g(con)o(v)o (erge.)26 b(Lev)o(en)o(b)q(erg-Marquardt)17 b(metho)q(ds)h(handle)h (the)e(inde\014nite)j(Hessian)e(matrix)0 333 y(problem;)c(ho)o(w)o(ev)o (er,)f(a)g(one-dimensional)j(optimization)f(or)e(other)g(form)g(of)g (searc)o(h)h(is)g(required)g(for)f(a)h(suitable)0 402 y(scalar)k(to)g(b)q(e)g(added)h(to)e(the)i(diagonal)f(elemen)o(ts)h(of) f(Hessian.)29 b(Fisher)19 b(scoring)f(metho)q(ds)g(can)g(also)h(handle) 0 471 y(the)13 b(inde\014nite)j(Hessian)e(matrix)f(problem,)h(but)g (for)f(non-quadratic)h(nonlinear)g(optimization)h(Fisher)f(scoring)0 539 y(requires)f(a)f(stepsize)h Fo(\021)h Fw(that)d(ob)q(eys)i Fm(k)p Fo(I)7 b Fw(+)t Fo(\021)r(B)r(H)t Fw(\(\002)868 523 y Fp(\()p Fn(k)q Fl(\000)p Fp(1\))962 539 y Fw(\)\))p Fm(k)12 b Fo(<)h Fw(1,)f(where)g Fo(B)j Fw(is)e(the)f(Fisher)h (information)f(matrix.)0 608 y(Th)o(us,)i(problems)g(similar)h(to)f (those)f(of)h(gradien)o(t)g(ascen)o(t)f(arise)h(here)h(as)e(w)o(ell.)21 b(Finally)l(,)15 b(for)e(the)h(quasi-Newton)0 677 y(metho)q(ds)f(or)f (conjugate)h(gradien)o(t)g(metho)q(ds,)g(a)g(one-dimensional)i(line)f (searc)o(h)f(is)h(required)g(at)e(eac)o(h)h(iteration.)0 746 y(In)h(summary)l(,)f(all)i(of)e(these)h(gradien)o(t-based)g(metho)q (ds)g(incur)g(extra)f(computational)h(costs)f(at)g(eac)o(h)h (iteration,)0 815 y(rendering)i(simple)h(comparisons)e(based)h(on)f(lo) q(cal)h(con)o(v)o(ergence)g(rates)e(unreliable.)71 884 y(F)l(or)i(large)h(scale)h(problems,)g(algorithms)f(that)g(c)o(hange)g (the)g(parameters)g(immediately)i(after)d(eac)o(h)h(data)0 953 y(p)q(oin)o(t)h(\(\\on-line)g(algorithms"\))f(are)g(often)g (signi\014can)o(tly)i(faster)d(in)i(practice)g(than)f(batc)o(h)g (algorithms.)27 b(The)0 1022 y(p)q(opularit)o(y)19 b(of)f(gradien)o(t)g (descen)o(t)h(algorithms)f(for)g(neural)h(net)o(w)o(orks)e(is)i(in)g (part)f(to)f(the)i(ease)f(of)g(obtaining)0 1090 y(on-line)g(v)m(arian)o (ts)f(of)f(gradien)o(t)h(descen)o(t.)24 b(It)17 b(is)g(w)o(orth)e (noting)i(that)f(on-line)i(v)m(arian)o(ts)f(of)f(the)h(EM)f(algorithm)0 1159 y(can)g(b)q(e)g(deriv)o(ed)h(\(Neal)e(&)h(Hin)o(ton,)g(1993,)e (Titterington,)i(1984\),)d(and)j(this)g(is)g(a)g(further)f(factor)g (that)g(w)o(eighs)0 1228 y(in)h(fa)o(v)o(or)e(of)h(EM)f(as)h(compared)g (to)g(conjugate)g(gradien)o(t)g(and)g(Newton)g(metho)q(ds.)0 1386 y Fq(5)69 b(Con)n(v)n(ergence)23 b(rate)g(comparisons)0 1500 y Fw(In)c(this)g(section,)h(w)o(e)e(pro)o(vide)i(a)e(comparativ)o (e)g(theoretical)h(discussion)i(of)d(the)h(con)o(v)o(ergence)f(rates)g (of)g(con-)0 1568 y(strained)e(gradien)o(t)f(ascen)o(t)g(and)g(EM.)71 1637 y(F)l(or)k(gradien)o(t)g(ascen)o(t)h(a)f(lo)q(cal)i(con)o(v)o (ergence)f(result)g(can)g(b)o(y)g(obtained)g(b)o(y)g(T)l(a)o(ylor)f (expanding)i(the)f(log)0 1706 y(lik)o(eliho)q(o)q(d)e(around)d(the)h (maxim)o(um)f(lik)o(eliho)q(o)q(d)j(estimate)d(\002)1076 1690 y Fl(\003)1096 1706 y Fw(.)20 b(F)l(or)14 b(su\016cien)o(tly)j (large)e Fo(k)h Fw(w)o(e)f(ha)o(v)o(e:)515 1807 y Fm(k)p Fw(\002)573 1788 y Fp(\()p Fn(k)q Fp(+1\))677 1807 y Fm(\000)10 b Fw(\002)757 1788 y Fl(\003)777 1807 y Fm(k)i(\024)h(k)p Fo(I)h Fw(+)c Fo(\021)r(H)t Fw(\(\002)1082 1788 y Fl(\003)1101 1807 y Fw(\)\))p Fm(kk)p Fw(\(\002)1236 1788 y Fp(\()p Fn(k)q Fp(\))1293 1807 y Fm(\000)h Fw(\002)1374 1788 y Fl(\003)1394 1807 y Fw(\))p Fm(k)433 b Fw(\(12\))0 1908 y(and)591 1977 y Fm(k)p Fo(I)13 b Fw(+)d Fo(\021)r(H)t Fw(\(\002)812 1958 y Fl(\003)831 1977 y Fw(\))p Fm(k)i(\024)h Fo(\025)959 1984 y Fn(M)998 1977 y Fw([)p Fo(I)g Fw(+)e Fo(\021)r(H)t Fw(\(\002)1210 1958 y Fl(\003)1228 1977 y Fw(\)])h(=)h Fo(r)o(;)516 b Fw(\(13\))0 2066 y(where)11 b Fo(H)j Fw(is)d(the)g(Hessian)h(of)e Fo(l)q Fw(,)h Fo(\021)g Fw(is)h(the)f(step)f(size,)i(and)f Fo(r)j Fw(=)f(max)o Fm(fj)p Fw(1)q Fm(\000)q Fo(\021)r(\025)1277 2073 y Fn(M)1315 2066 y Fw([)p Fm(\000)p Fo(H)t Fw(\(\002)1458 2049 y Fl(\003)1477 2066 y Fw(\)])p Fm(j)p Fo(;)k Fm(j)p Fw(1)q Fm(\000)q Fo(\021)r(\025)1676 2073 y Fn(m)1708 2066 y Fw([)p Fm(\000)p Fo(H)t Fw(\(\002)1851 2049 y Fl(\003)1871 2066 y Fw(\)])p Fm(jg)p Fw(,)0 2135 y(where)e Fo(\025)158 2142 y Fn(M)197 2135 y Fw([)p Fo(A)p Fw(])g(and)g Fo(\025)387 2142 y Fn(m)420 2135 y Fw([)p Fo(A)p Fw(])g(denote)g(the)g(largest)g (and)h(smallest)f(eigen)o(v)m(alues)i(of)e Fo(A)p Fw(,)g(resp)q(ectiv)o (ely)l(.)71 2203 y(Smaller)23 b(v)m(alues)h(of)e Fo(r)i Fw(corresp)q(ond)f(to)f(faster)f(con)o(v)o(ergence)i(rates.)42 b(T)l(o)22 b(guaran)o(tee)g(con)o(v)o(ergence,)i(w)o(e)0 2272 y(require)c Fo(r)h(<)f Fw(1)g(or)f(0)g Fo(<)i(\021)g(<)f Fw(2)p Fo(=\025)627 2279 y Fn(M)665 2272 y Fw([)p Fm(\000)p Fo(H)t Fw(\(\002)808 2256 y Fl(\003)828 2272 y Fw(\)].)32 b(The)20 b(minim)o(um)g(p)q(ossible)i(v)m(alue)f(of)e Fo(r)h Fw(is)g(obtained)g(when)0 2341 y Fo(\021)14 b Fw(=)f(1)p Fo(=\025)158 2348 y Fn(M)196 2341 y Fw([)p Fo(H)t Fw(\(\002)304 2325 y Fl(\003)323 2341 y Fw(\)])i(with)601 2442 y Fo(r)622 2449 y Fn(min)730 2442 y Fw(=)42 b(1)9 b Fm(\000)i Fo(\025)912 2449 y Fn(m)945 2442 y Fw([)p Fo(H)t Fw(\(\002)1053 2423 y Fl(\003)1072 2442 y Fw(\)])p Fo(=\025)1153 2449 y Fn(M)1191 2442 y Fw([)p Fo(H)t Fw(\(\002)1299 2423 y Fl(\003)1317 2442 y Fw(\)])730 2523 y Fm(\021)42 b Fw(1)9 b Fm(\000)i Fo(\024)911 2505 y Fl(\000)p Fp(1)958 2523 y Fw([)p Fo(H)t Fw(\(\002)1066 2505 y Fl(\003)1085 2523 y Fw(\)])p Fo(;)0 2624 y Fw(where)k Fo(\024)p Fw([)p Fo(H)t Fw(])d(=)h Fo(\025)312 2631 y Fn(M)351 2624 y Fw([)p Fo(H)t Fw(])p Fo(=\025)469 2631 y Fn(m)500 2624 y Fw([)p Fo(H)t Fw(])h(is)h(the)g Fx(c)n(ondition)h(numb)n(er)e Fw(of)h Fo(H)t Fw(.)k(Larger)c(v)m(alues)h(of)e(the)h(condition)h(n)o (um)o(b)q(er)0 2693 y(corresp)q(ond)21 b(to)e(slo)o(w)o(er)h(con)o(v)o (ergence.)35 b(When)20 b Fo(\024)p Fw([)p Fo(H)t Fw(])g(=)h(1)f(w)o(e)g (ha)o(v)o(e)g Fo(r)1276 2700 y Fn(min)1363 2693 y Fw(=)h(0,)g(whic)o(h) g(corresp)q(onds)g(to)e(a)963 2817 y(9)p eop %%Page: 10 10 10 9 bop 0 195 a Fw(sup)q(erlinear)15 b(rate)d(of)g(con)o(v)o(ergence.) 19 b(Indeed,)14 b(Newton's)e(metho)q(d)h(can)g(b)q(e)g(view)o(ed)h(as)e (a)g(metho)q(d)h(for)f(obtaining)0 264 y(a)18 b(more)h(desirable)h (condition)g(n)o(um)o(b)q(er|the)f(in)o(v)o(erse)g(Hessian)h Fo(H)1198 247 y Fl(\000)p Fp(1)1263 264 y Fw(balances)f(the)g(Hessian)g Fo(H)k Fw(suc)o(h)18 b(that)0 333 y(the)c(resulting)h(condition)g(n)o (um)o(b)q(er)f(is)h(one.)k(E\013ectiv)o(ely)l(,)c(Newton)f(can)g(b)q(e) g(regarded)g(as)g(gradien)o(t)g(ascen)o(t)f(on)h(a)0 402 y(new)g(function)g(with)g(an)g(e\013ectiv)o(e)g(Hessian)g(that)f (is)i(the)e(iden)o(tit)o(y)i(matrix:)k Fo(H)1358 409 y Fn(ef)t(f)1430 402 y Fw(=)13 b Fo(H)1520 385 y Fl(\000)p Fp(1)1566 402 y Fo(H)j Fw(=)d Fo(I)t Fw(.)19 b(In)c(practice,)0 471 y(ho)o(w)o(ev)o(er,)i Fo(\024)p Fw([)p Fo(H)t Fw(])f(is)i(usually)g (quite)g(large.)27 b(The)17 b(larger)g Fo(\024)p Fw([)p Fo(H)t Fw(])g(is,)h(the)f(more)g(di\016cult)i(it)e(is)h(to)f(compute)g Fo(H)1902 454 y Fl(\000)p Fp(1)0 539 y Fw(accurately)l(.)k(Hence)16 b(it)f(is)h(di\016cult)h(to)d(balance)j(the)e(Hessian)h(as)f(desired.) 21 b(In)16 b(addition,)g(as)f(w)o(e)g(men)o(tioned)h(in)0 608 y(the)i(previous)g(section,)g(the)g(Hessian)g(v)m(aries)g(from)f(p) q(oin)o(t)h(to)f(p)q(oin)o(t)h(in)h(the)e(parameter)g(space,)h(and)g (at)f(eac)o(h)0 677 y(iteration)f(w)o(e)f(need)i(recompute)f(the)f(in)o (v)o(erse)h(Hessian.)22 b(Quasi-Newton)17 b(metho)q(ds)e(appro)o (ximate)h Fo(H)t Fw(\(\002)1836 661 y Fp(\()p Fn(k)q Fp(\))1884 677 y Fw(\))1902 661 y Fl(\000)p Fp(1)0 746 y Fw(b)o(y)f(a)g(p)q(ositiv)o(e)h(matrix)f Fo(B)453 730 y Fp(\()p Fn(k)q Fp(\))518 746 y Fw(that)f(is)i(easy)f(to)f(compute.)71 815 y(The)i(discussion)i(th)o(us)d(far)h(has)g(treated)f(unconstrained) j(optimization.)23 b(In)17 b(order)f(to)f(compare)h(gradien)o(t)0 884 y(ascen)o(t)22 b(with)i(the)e(EM)h(algorithm)g(on)f(the)h (constrained)h(mixture)f(estimation)g(problem,)i(w)o(e)d(consider)i(a)0 953 y(gradien)o(t)15 b(pro)s(jection)g(metho)q(d:)709 1028 y(\002)744 1009 y Fp(\()p Fn(k)q Fp(+1\))851 1028 y Fw(=)e(\002)934 1009 y Fp(\()p Fn(k)q Fp(\))994 1028 y Fw(+)d Fo(\021)r Fw(\005)1098 1035 y Fn(k)1158 997 y Fo(@)s(l)p 1124 1017 111 2 v 1124 1061 a(@)s Fw(\002)1186 1047 y Fp(\()p Fn(k)q Fp(\))1868 1028 y Fw(\(14\))0 1127 y(where)i(\005)162 1134 y Fn(k)195 1127 y Fw(is)h(the)f(pro)s(jection)f (matrix)h(that)f(pro)s(jects)g(the)h(gradien)o(t)1219 1109 y Fn(@)r(l)p 1190 1116 92 2 v 1190 1146 a(@)r Fp(\002)1238 1137 y Fj(\()p Ff(k)q Fj(\))1297 1127 y Fw(in)o(to)g Fm(D)1421 1134 y Fp(1)1441 1127 y Fw(.)19 b(This)12 b(gradien)o(t)g (pro)s(jection)0 1196 y(iteration)17 b(will)h(remain)f(in)g Fm(D)515 1203 y Fp(1)551 1196 y Fw(as)g(long)f(as)g(the)h(initial)i (parameter)c(v)o(ector)h(is)h(in)g Fm(D)1470 1203 y Fp(1)1490 1196 y Fw(.)24 b(T)l(o)16 b(k)o(eep)h(the)g(iteration)0 1265 y(within)f Fm(D)176 1272 y Fp(2)196 1265 y Fw(,)f(w)o(e)g(c)o(ho)q (ose)g(an)g(initial)j(\002)665 1248 y Fp(\(0\))725 1265 y Fm(2)13 b(D)803 1272 y Fp(2)838 1265 y Fw(and)i(k)o(eep)g Fo(\021)i Fw(su\016cien)o(tly)f(small)g(at)f(eac)o(h)g(iteration.)71 1333 y(Supp)q(ose)i(that)f Fo(E)h Fw(=)e([)p Fo(e)485 1340 y Fp(1)505 1333 y Fo(;)8 b Fm(\001)g(\001)g(\001)t Fo(;)g(e)627 1340 y Fn(m)660 1333 y Fw(])16 b(are)g(a)g(set)h(of)f (indep)q(enden)o(t)j(unit)e(basis)g(v)o(ectors)f(that)f(span)i(the)g (space)0 1402 y Fm(D)35 1409 y Fp(1)55 1402 y Fw(.)29 b(In)18 b(this)h(basis,)g(\002)411 1386 y Fp(\()p Fn(k)q Fp(\))478 1402 y Fw(and)g(\005)604 1409 y Fn(k)660 1384 y(@)r(l)p 630 1391 V 630 1422 a(@)r Fp(\002)678 1412 y Fj(\()p Ff(k)q Fj(\))744 1402 y Fw(b)q(ecome)g(\002)945 1378 y Fp(\()p Fn(k)q Fp(\))945 1408 y Fn(c)1012 1402 y Fw(=)f Fo(E)1102 1386 y Fn(T)1128 1402 y Fw(\002)1163 1386 y Fp(\()p Fn(k)q Fp(\))1231 1402 y Fw(and)1357 1384 y Fn(@)r(l)p 1327 1391 V 1327 1422 a(@)r Fp(\002)1375 1412 y Fj(\()p Ff(k)q Fj(\))1423 1424 y Fn(c)1458 1402 y Fw(=)g Fo(E)1548 1386 y Fn(T)1610 1384 y(@)r(l)p 1580 1391 V 1580 1422 a(@)r Fp(\002)1628 1412 y Fj(\()p Ff(k)q Fj(\))1676 1402 y Fw(,)g(resp)q(ectiv)o(ely)l(,)0 1477 y(with)j Fm(k)p Fw(\002)167 1453 y Fp(\()p Fn(k)q Fp(\))167 1482 y Fn(c)229 1477 y Fm(\000)14 b Fw(\002)313 1461 y Fl(\003)313 1488 y Fn(c)333 1477 y Fm(k)21 b Fw(=)g Fm(k)p Fw(\002)491 1461 y Fp(\()p Fn(k)q Fp(\))554 1477 y Fm(\000)13 b Fw(\002)637 1461 y Fl(\003)657 1477 y Fm(k)p Fw(.)35 b(In)21 b(this)g(represen)o(tation)f(the)h(pro)s(jectiv) o(e)f(gradien)o(t)g(algorithm)g(Eq.)0 1546 y(\(14\))i(b)q(ecomes)h (simple)i(gradien)o(t)e(ascen)o(t:)35 b(\002)834 1522 y Fp(\()p Fn(k)q Fp(+1\))834 1551 y Fn(c)954 1546 y Fw(=)26 b(\002)1050 1522 y Fp(\()p Fn(k)q Fp(\))1050 1551 y Fn(c)1114 1546 y Fw(+)16 b Fo(\021)1224 1528 y Fn(@)r(l)p 1194 1535 V 1194 1573 a(@)r Fp(\002)1242 1556 y Fj(\()p Ff(k)q Fj(\))1242 1579 y Ff(c)1290 1546 y Fw(.)43 b(Moreo)o(v)o(er,)23 b(Eq.)43 b(\(12\))22 b(b)q(ecomes)0 1625 y Fm(k)p Fw(\002)58 1608 y Fp(\()p Fn(k)q Fp(+1\))162 1625 y Fm(\000)10 b Fw(\002)242 1608 y Fl(\003)263 1625 y Fm(k)15 b(\024)h(k)p Fo(E)412 1608 y Fn(T)439 1625 y Fw(\()p Fo(I)e Fw(+)e Fo(\021)r(H)t Fw(\(\002)658 1608 y Fl(\003)676 1625 y Fw(\)\))p Fm(kk)p Fw(\(\002)811 1608 y Fp(\()p Fn(k)q Fp(\))870 1625 y Fm(\000)g Fw(\002)952 1608 y Fl(\003)972 1625 y Fw(\))p Fm(k)p Fw(.)25 b(As)17 b(a)g(result,)h(the)f(con)o(v)o (ergence)g(rate)g(is)g(b)q(ounded)0 1694 y(b)o(y)496 1797 y Fo(r)517 1804 y Fn(c)575 1797 y Fw(=)42 b Fm(k)p Fo(E)712 1778 y Fn(T)738 1797 y Fw(\()p Fo(I)14 b Fw(+)c Fo(\021)r(H)t Fw(\(\002)955 1778 y Fl(\003)974 1797 y Fw(\)\))p Fm(k)575 1878 y(\024)652 1826 y Fe(q)p 693 1826 760 2 v 693 1878 a Fo(\025)720 1885 y Fn(M)759 1878 y Fw([)p Fo(E)809 1865 y Fn(T)836 1878 y Fw(\()p Fo(I)j Fw(+)e Fo(\021)r(H)t Fw(\(\002)1053 1865 y Fl(\003)1071 1878 y Fw(\)\)\()p Fo(I)i Fw(+)d Fo(\021)r(H)t Fw(\(\002)1323 1865 y Fl(\003)1341 1878 y Fw(\)\))1377 1865 y Fn(T)1404 1878 y Fo(E)s Fw(])575 1971 y(=)652 1919 y Fe(q)p 693 1919 736 2 v 693 1971 a Fo(\025)720 1978 y Fn(M)759 1971 y Fw([)p Fo(E)809 1958 y Fn(T)836 1971 y Fw(\()p Fo(I)j Fw(+)e(2)p Fo(\021)r(H)t Fw(\(\002)1076 1958 y Fl(\003)1094 1971 y Fw(\))e(+)i Fo(\021)1192 1958 y Fp(2)1211 1971 y Fo(H)1253 1958 y Fp(2)1272 1971 y Fw(\(\002)1325 1958 y Fl(\003)1345 1971 y Fw(\)\))p Fo(E)s Fw(])n Fo(:)0 2074 y Fw(Since)17 b Fo(H)t Fw(\(\002)214 2057 y Fl(\003)233 2074 y Fw(\))e(is)g(negativ)o(e)h(de\014nite,)g(w)o(e)f(obtain)607 2177 y Fo(r)628 2184 y Fn(c)658 2177 y Fm(\024)706 2125 y Fe(q)p 747 2125 582 2 v 747 2177 a Fw(1)10 b(+)g Fo(\021)850 2164 y Fp(2)869 2177 y Fo(\025)896 2161 y Fp(2)896 2190 y Fn(M)935 2177 y Fw([)p Fm(\000)p Fo(H)1021 2184 y Fn(c)1038 2177 y Fw(])g Fm(\000)h Fw(2)p Fo(\021)r(\025)1182 2184 y Fn(m)1213 2177 y Fw([)p Fm(\000)p Fo(H)1299 2184 y Fn(c)1316 2177 y Fw(])p Fo(:)526 b Fw(\(15\))0 2280 y(In)16 b(this)f(equation)h Fo(H)369 2287 y Fn(c)399 2280 y Fw(=)d Fo(E)484 2263 y Fn(T)511 2280 y Fo(H)t Fw(\(\002\))p Fo(E)j Fw(is)g(the)f(Hessian)h(of)f Fo(l)g Fw(restricted)h(to)e Fm(D)1338 2287 y Fp(1)1358 2280 y Fw(.)71 2349 y(W)l(e)e(see)g(from)f (this)i(deriv)m(ation)g(that)e(the)h(con)o(v)o(ergence)g(sp)q(eed)h (dep)q(ends)h(on)e Fo(\024)p Fw([)p Fo(H)1466 2356 y Fn(c)1483 2349 y Fw(])g(=)h Fo(\025)1583 2356 y Fn(M)1622 2349 y Fw([)p Fm(\000)p Fo(H)1708 2356 y Fn(c)1725 2349 y Fw(])p Fo(=\025)1788 2356 y Fn(m)1820 2349 y Fw([)p Fm(\000)p Fo(H)1906 2356 y Fn(c)1924 2349 y Fw(].)0 2417 y(When)k Fo(\024)p Fw([)p Fo(H)211 2424 y Fn(c)228 2417 y Fw(])d(=)h(1,)i(w)o(e)f(ha)o(v)o(e)532 2368 y Fe(q)p 574 2368 593 2 v 49 x Fw(1)9 b(+)i Fo(\021)677 2404 y Fp(2)696 2417 y Fo(\025)723 2402 y Fp(2)723 2431 y Fn(M)762 2417 y Fw(\()p Fm(\000)p Fo(H)853 2424 y Fn(c)870 2417 y Fw(\))f Fm(\000)g Fw(2)p Fo(\021)r(\025)1018 2424 y Fn(m)1050 2417 y Fw([)p Fm(\000)p Fo(H)1136 2424 y Fn(c)1153 2417 y Fw(])k(=)i(1)10 b Fm(\000)i Fo(\021)r(\025)p Fw([)p Fm(\000)p Fo(H)1449 2424 y Fn(c)1465 2417 y Fw(],)k(whic)o(h)h(in)g (principle)j(can)0 2486 y(b)q(e)f(made)f(to)f(equal)h(zero)g(if)h Fo(\021)g Fw(is)f(selected)h(appropriately)l(.)29 b(In)19 b(this)f(case,)g(a)g(sup)q(erlinear)i(rate)d(is)h(obtained.)0 2555 y(Generally)l(,)e(ho)o(w)o(ev)o(er,)e Fo(\024)p Fw([)p Fo(H)479 2562 y Fn(c)496 2555 y Fw(])e Fm(6)p Fw(=)h(1,)i(with)g(smaller)h(v)m(alues)h(of)d Fo(\024)p Fw([)p Fo(H)1144 2562 y Fn(c)1161 2555 y Fw(])h(corresp)q(onding)h(to)f (faster)f(con)o(v)o(ergence.)71 2624 y(W)l(e)j(no)o(w)g(turn)h(to)f(an) g(analysis)h(of)f(the)h(EM)f(algorithm.)27 b(As)18 b(w)o(e)f(ha)o(v)o (e)g(seen)h(EM)f(k)o(eeps)h(the)g(parameter)0 2693 y(v)o(ector)d (within)j Fm(D)315 2700 y Fp(1)351 2693 y Fw(automatically)l(.)23 b(Th)o(us,)16 b(in)h(the)g(new)f(basis)h(the)f(connection)h(b)q(et)o(w) o(een)g(EM)e(and)i(gradien)o(t)952 2817 y(10)p eop %%Page: 11 11 11 10 bop 0 195 a Fw(ascen)o(t)15 b(\(cf.)k(Eq.)h(\(10\)\))14 b(b)q(ecomes)664 319 y(\002)699 300 y Fp(\()p Fn(k)q Fp(+1\))699 330 y Fn(c)806 319 y Fw(=)f(\002)889 300 y Fp(\()p Fn(k)q Fp(\))889 330 y Fn(c)948 319 y Fw(+)e Fo(E)1031 300 y Fn(T)1058 319 y Fo(P)6 b Fw(\(\002)1146 300 y Fp(\()p Fn(k)q Fp(\))1195 319 y Fw(\))1228 288 y Fo(@)s(l)p 1218 308 63 2 v 1218 350 a(@)s Fw(\002)0 433 y(and)15 b(w)o(e)g(ha)o(v)o(e)468 502 y Fm(k)p Fw(\002)526 483 y Fp(\()p Fn(k)q Fp(+1\))630 502 y Fm(\000)c Fw(\002)711 483 y Fl(\003)731 502 y Fm(k)h(\024)h(k)p Fo(E)874 483 y Fn(T)900 502 y Fw(\()p Fo(I)h Fw(+)c Fo(P)c(H)t Fw(\(\002)1127 483 y Fl(\003)1147 502 y Fw(\)\))p Fm(kk)p Fw(\(\002)1282 483 y Fp(\()p Fn(k)q Fp(\))1340 502 y Fm(\000)k Fw(\002)1420 483 y Fl(\003)1440 502 y Fw(\))p Fm(k)0 598 y Fw(with)281 667 y Fo(r)302 674 y Fn(c)332 667 y Fw(=)j Fm(k)p Fo(E)440 648 y Fn(T)466 667 y Fw(\()p Fo(I)g Fw(+)e Fo(P)6 b(H)t Fw(\(\002)693 648 y Fl(\003)713 667 y Fw(\)\))p Fm(k)11 b(\024)831 615 y Fe(q)p 873 615 783 2 v 52 x Fo(\025)900 674 y Fn(M)939 667 y Fw([)p Fo(E)989 654 y Fn(T)1015 667 y Fw(\()p Fo(I)i Fw(+)e Fo(P)6 b(H)t Fw(\(\002)1242 654 y Fl(\003)1262 667 y Fw(\)\)\()p Fo(I)12 b Fw(+)f Fo(P)6 b(H)t Fw(\(\002)1524 654 y Fl(\003)1543 667 y Fw(\)\))1579 654 y Fn(T)1606 667 y Fo(E)s Fw(])o Fo(:)0 763 y Fw(The)15 b(latter)g(equation)h(can)f(b)q(e)h(further)f (manipulated)i(to)d(yield:)537 877 y Fo(r)558 884 y Fn(c)588 877 y Fm(\024)636 826 y Fe(q)p 677 826 722 2 v 677 877 a Fw(1)c(+)g Fo(\025)782 861 y Fp(2)782 891 y Fn(M)821 877 y Fw([)p Fo(E)871 864 y Fn(T)898 877 y Fo(P)c(H)t(E)s Fw(])j Fm(\000)h Fw(2)p Fo(\025)1129 884 y Fn(m)1162 877 y Fw([)p Fm(\000)p Fo(E)1247 864 y Fn(T)1273 877 y Fo(P)c(H)t(E)s Fw(])o Fo(:)456 b Fw(\(16\))0 992 y(Th)o(us)10 b(w)o(e)g(see)g(that)g(the)g(con)o(v)o(ergence)g(sp)q(eed)i(of)d(EM)h (dep)q(ends)i(on)e Fo(\024)p Fw([)p Fo(E)1208 975 y Fn(T)1234 992 y Fo(P)c(H)t(E)s Fw(])12 b(=)h Fo(\025)1448 999 y Fn(M)1487 992 y Fw([)p Fo(E)1537 975 y Fn(T)1563 992 y Fo(P)6 b(H)t(E)s Fw(])p Fo(=\025)1740 999 y Fn(m)1771 992 y Fw([)p Fo(E)1821 975 y Fn(T)1848 992 y Fo(P)g(H)t(E)s Fw(].)0 1061 y(When)24 b Fo(\024)p Fw([)p Fo(E)217 1044 y Fn(T)244 1061 y Fo(P)6 b(H)t(E)s Fw(])25 b(=)i(1,)e Fo(\025)546 1068 y Fn(M)585 1061 y Fw([)p Fo(E)635 1044 y Fn(T)662 1061 y Fo(P)6 b(H)t(E)s Fw(])25 b(=)j(1,)d(w)o(e)e(ha)o(v)o (e)1126 1011 y Fe(q)p 1168 1011 V 50 x Fw(1)10 b(+)g Fo(\025)1273 1045 y Fp(2)1273 1074 y Fn(M)1312 1061 y Fw([)p Fo(E)1362 1047 y Fn(T)1389 1061 y Fo(P)c(H)t(E)s Fw(])i Fm(\000)j Fw(2)p Fo(\025)1620 1068 y Fn(m)1652 1061 y Fw([)p Fm(\000)p Fo(E)1737 1047 y Fn(T)1764 1061 y Fo(P)6 b(H)t(E)s Fw(])22 b(=)0 1129 y(\(1)7 b Fm(\000)i Fo(\025)119 1136 y Fn(M)158 1129 y Fw([)p Fm(\000)p Fo(E)243 1113 y Fn(T)269 1129 y Fo(P)d(H)t(E)s Fw(]\))11 b(=)i(0.)20 b(In)14 b(this)h(case,)f(a)g(sup)q(erlinear)i(rate)d(is)i(obtained.)20 b(W)l(e)14 b(discuss)i(the)e(p)q(ossibilit)o(y)i(of)0 1198 y(obtaining)g(sup)q(erlinear)h(con)o(v)o(ergence)f(with)f(EM)g(in) h(more)f(detail)h(b)q(elo)o(w.)71 1267 y(These)c(results)g(sho)o(w)f (that)g(the)h(con)o(v)o(ergence)g(of)g(gradien)o(t)g(ascen)o(t)f(and)h (EM)g(b)q(oth)g(dep)q(end)h(on)f(the)g(shap)q(e)g(of)0 1336 y(the)i(log)h(lik)o(eliho)q(o)q(d)i(as)c(measured)i(b)o(y)f(the)g (condition)i(n)o(um)o(b)q(er.)k(When)14 b Fo(\024)p Fw([)p Fo(H)t Fw(])f(is)i(near)f(one,)g(the)h(con\014guration)0 1405 y(is)h(quite)f(regular,)g(and)g(the)g(up)q(date)h(direction)g(p)q (oin)o(ts)g(directly)g(to)f(the)g(solution)g(yielding)j(fast)c(con)o(v) o(ergence.)0 1474 y(When)k Fo(\024)p Fw([)p Fo(H)t Fw(])f(is)h(v)o(ery) g(large,)g(the)g Fo(l)g Fw(surface)g(has)g(an)g(elongated)g(shap)q(e,)h (and)f(the)g(searc)o(h)f(along)h(the)g(up)q(date)0 1543 y(direction)12 b(is)g(a)f(zigzag)g(path,)g(making)g(con)o(v)o(ergence)h (v)o(ery)e(slo)o(w.)19 b(The)11 b(k)o(ey)g(idea)h(of)e(Newton)h(and)g (quasi-Newton)0 1612 y(metho)q(ds)j(is)h(to)f(reshap)q(e)h(the)f (surface.)20 b(The)14 b(nearer)g(it)h(is)g(to)e(a)h(ball)i(shap)q(e)f (\(Newton's)e(metho)q(d)h(ac)o(hiev)o(es)h(this)0 1680 y(shap)q(e)20 b(in)g(the)f(ideal)h(case\),)g(the)f(b)q(etter)g(the)g (con)o(v)o(ergence.)32 b(Quasi-Newton)20 b(metho)q(ds)f(aim)h(to)e(ac)o (hiev)o(e)i(an)0 1749 y(e\013ectiv)o(e)e(Hessian)g(whose)f(condition)h (n)o(um)o(b)q(er)g(is)g(as)f(close)h(as)e(p)q(ossible)j(to)e(one.)26 b(In)o(terestingly)l(,)19 b(the)e(results)0 1818 y(that)i(w)o(e)h(no)o (w)f(presen)o(t)h(suggest)f(that)g(the)h(pro)s(jection)f(matrix)h Fo(P)26 b Fw(for)19 b(the)h(EM)f(algorithm)h(also)g(serv)o(es)f(to)0 1887 y(e\013ectiv)o(ely)14 b(reshap)q(e)h(the)e(lik)o(eliho)q(o)q(d)k (yielding)e(an)f(e\013ectiv)o(e)g(condition)h(n)o(um)o(b)q(er)f(that)e (tends)i(to)f(one.)20 b(W)l(e)13 b(\014rst)0 1956 y(presen)o(t)i (empirical)i(results)f(that)e(supp)q(ort)i(this)f(suggestion)h(and)f (then)g(presen)o(t)h(a)f(theoretical)g(analysis.)71 2025 y(W)l(e)g(sampled)h(1000)e(p)q(oin)o(ts)i(from)e(a)h(simple)i(\014nite) f(mixture)f(mo)q(del)h(giv)o(en)g(b)o(y)722 2139 y Fo(p)p Fw(\()p Fo(x)p Fw(\))11 b(=)i Fo(\013)895 2146 y Fp(1)915 2139 y Fo(p)938 2146 y Fp(1)958 2139 y Fw(\()p Fo(x)p Fw(\))c(+)i Fo(\013)1104 2146 y Fp(2)1123 2139 y Fo(p)1146 2146 y Fp(2)1166 2139 y Fw(\()p Fo(x)p Fw(\))0 2254 y(where)619 2323 y Fo(p)642 2330 y Fn(i)656 2323 y Fw(\()p Fo(x)p Fw(\))g(=)841 2292 y(1)p 782 2312 140 2 v 782 2321 a Fe(q)p 824 2321 98 2 v 49 x Fw(2)p Fo(\031)r(\033)903 2354 y Fp(2)901 2383 y Fn(i)934 2323 y Fw(exp)q Fm(f\000)1067 2292 y Fw(1)p 1067 2312 23 2 v 1067 2354 a(2)1099 2292 y(\()p Fo(x)f Fm(\000)h Fo(m)1239 2299 y Fn(i)1253 2292 y Fw(\))1271 2276 y Fp(2)p 1099 2312 191 2 v 1171 2354 a Fo(\033)1199 2339 y Fp(2)1197 2367 y Fn(i)1295 2323 y Fm(g)p Fo(:)0 2463 y Fw(The)21 b(parameter)f(v)m(alues)i(w)o(ere)f (as)f(follo)o(ws:)31 b Fo(\013)842 2470 y Fp(1)884 2463 y Fw(=)22 b(0)p Fo(:)p Fw(7170)p Fo(;)k(\013)1137 2470 y Fp(2)1179 2463 y Fw(=)c(0)p Fo(:)p Fw(2830)p Fo(;)27 b(m)1444 2470 y Fp(1)1485 2463 y Fw(=)c Fm(\000)p Fw(2)p Fo(;)28 b(m)1682 2470 y Fp(2)1723 2463 y Fw(=)22 b(2)p Fo(;)28 b(\033)1872 2446 y Fp(2)1870 2474 y(1)1913 2463 y Fw(=)0 2531 y(1)p Fo(;)c(\033)88 2515 y Fp(2)86 2543 y(2)122 2531 y Fw(=)16 b(1.)25 b(W)l(e)16 b(ran)h(b)q(oth)g(the)g(EM)f (algorithm)h(and)g(gradien)o(t)g(ascen)o(t)f(on)h(the)g(data.)24 b(A)o(t)17 b(eac)o(h)f(step)h(of)g(the)0 2600 y(sim)o(ulation,)e(w)o(e) f(calculated)i(the)f(condition)g(n)o(um)o(b)q(er)g(of)f(the)g(Hessian)h (\()p Fo(\024)p Fw([)p Fo(H)t Fw(\(\002)1399 2584 y Fp(\()p Fn(k)q Fp(\))1447 2600 y Fw(\)]\),)e(the)i(condition)g(n)o(um)o(b)q(er) 0 2669 y(determining)g(the)e(rate)g(of)g(con)o(v)o(ergence)h(of)f(the)g (gradien)o(t)h(algorithm)f(\()p Fo(\024)p Fw([)p Fo(E)1321 2653 y Fn(T)1347 2669 y Fo(H)t Fw(\(\002)1442 2653 y Fp(\()p Fn(k)q Fp(\))1491 2669 y Fw(\))p Fo(E)s Fw(]\),)e(and)j(the)f (condition)952 2817 y(11)p eop %%Page: 12 12 12 11 bop 0 195 a Fw(n)o(um)o(b)q(er)16 b(determining)i(the)e(rate)g (of)f(con)o(v)o(ergence)i(of)e(EM)h(\()p Fo(\024)p Fw([)p Fo(E)1134 179 y Fn(T)1160 195 y Fo(P)6 b Fw(\(\002)1248 179 y Fp(\()p Fn(k)q Fp(\))1297 195 y Fw(\))p Fo(H)t Fw(\(\002)1410 179 y Fp(\()p Fn(k)q Fp(\))1458 195 y Fw(\))p Fo(E)s Fw(]\).)21 b(W)l(e)16 b(also)g(calculated)0 264 y(the)22 b(largest)g(eigen)o(v)m(alues)h(of)f(the)g(matrices)g Fo(H)t Fw(\(\002)908 247 y Fp(\()p Fn(k)q Fp(\))956 264 y Fw(\),)h Fo(E)1047 247 y Fn(T)1073 264 y Fo(H)t Fw(\(\002)1168 247 y Fp(\()p Fn(k)q Fp(\))1217 264 y Fw(\))p Fo(E)s Fw(,)f(and)g Fo(E)1439 247 y Fn(T)1466 264 y Fo(P)6 b Fw(\(\002)1554 247 y Fp(\()p Fn(k)q Fp(\))1603 264 y Fw(\))p Fo(H)t Fw(\(\002)1716 247 y Fp(\()p Fn(k)q Fp(\))1764 264 y Fw(\))p Fo(E)s Fw(.)39 b(The)0 333 y(results)11 b(are)g(sho)o(wn)g(in)h(Fig.)18 b(1.)g(As)11 b(can)h(b)q(e)f(seen)h(in) g(Fig.)18 b(1\(a\),)11 b(the)g(condition)h(n)o(um)o(b)q(ers)f(c)o (hange)h(rapidly)g(in)g(the)0 402 y(vicinit)o(y)i(of)e(the)g(25th)g (iteration)g(and)h(the)f(corresp)q(onding)i(Hessian)f(matrices)f(b)q (ecome)h(inde\014nite.)21 b(Afterw)o(ard,)0 471 y(the)14 b(Hessians)g(quic)o(kly)h(b)q(ecome)g(de\014nite)g(and)f(the)f (condition)i(n)o(um)o(b)q(ers)f(con)o(v)o(erge.)1463 454 y Fp(3)1502 471 y Fw(As)g(sho)o(wn)f(in)i(Fig.)k(1\(b\),)0 539 y(the)c(condition)i(n)o(um)o(b)q(ers)e(con)o(v)o(erge)g(to)o(w)o (ard)f(the)h(v)m(alues)h Fo(\024)p Fw([)p Fo(H)t Fw(\(\002)1146 523 y Fp(\()p Fn(k)q Fp(\))1194 539 y Fw(\)])c(=)h(47)p Fo(:)p Fw(5,)h Fo(\024)p Fw([)p Fo(E)1470 523 y Fn(T)1497 539 y Fo(H)t Fw(\(\002)1592 523 y Fp(\()p Fn(k)q Fp(\))1640 539 y Fw(\))p Fo(E)s Fw(])d(=)i(33)p Fo(:)p Fw(5,)h(and)0 608 y Fo(\024)p Fw([)p Fo(E)76 592 y Fn(T)103 608 y Fo(P)6 b Fw(\(\002)191 592 y Fp(\()p Fn(k)q Fp(\))240 608 y Fw(\))p Fo(H)t Fw(\(\002)353 592 y Fp(\()p Fn(k)q Fp(\))401 608 y Fw(\))p Fo(E)s Fw(])15 b(=)h(3)p Fo(:)p Fw(6.)26 b(That)17 b(is,)h(the)g(matrix)f Fo(P)24 b Fw(has)17 b(greatly)g(reduced)i(the)e(condition)i(n)o(um)o(b)q(er,)0 677 y(b)o(y)c(factors)f(of)h(9)g(and)g(15.)20 b(This)15 b(signi\014can)o(tly)i(impro)o(v)o(es)e(the)g(shap)q(e)h(of)f Fo(l)h Fw(and)f(sp)q(eeds)h(up)g(the)f(con)o(v)o(ergence.)71 746 y(W)l(e)i(ran)g(a)g(second)h(exp)q(erimen)o(t)h(in)f(whic)o(h)g (the)g(means)f(of)g(the)h(comp)q(onen)o(t)f(Gaussians)h(w)o(ere)f Fo(m)1803 753 y Fp(1)1839 746 y Fw(=)g Fm(\000)p Fw(1)0 815 y(and)i Fo(m)132 822 y Fp(2)169 815 y Fw(=)g(1.)29 b(The)19 b(results)g(are)f(similar)h(to)f(those)g(sho)o(wn)g(in)i(Fig.) 29 b(1.)h(Since)20 b(the)e(distance)h(b)q(et)o(w)o(een)g(t)o(w)o(o)0 884 y(distributions)h(is)e(reduced)h(in)o(to)f(half,)h(the)f(shap)q(e)h (of)e Fo(l)i Fw(b)q(ecomes)f(more)g(irregular.)29 b(The)18 b(condition)h(n)o(um)o(b)q(er)0 953 y Fo(\024)p Fw([)p Fo(H)t Fw(\(\002)134 936 y Fp(\()p Fn(k)q Fp(\))182 953 y Fw(\)])11 b(increases)h(to)f(352,)g Fo(\024)p Fw([)p Fo(E)631 936 y Fn(T)658 953 y Fo(H)t Fw(\(\002)753 936 y Fp(\()p Fn(k)q Fp(\))801 953 y Fw(\))p Fo(E)s Fw(])f(increases)i(to)f (216,)g(and)h Fo(\024)p Fw([)p Fo(E)1371 936 y Fn(T)1398 953 y Fo(P)6 b Fw(\(\002)1486 936 y Fp(\()p Fn(k)q Fp(\))1535 953 y Fw(\))p Fo(H)t Fw(\(\002)1648 936 y Fp(\()p Fn(k)q Fp(\))1696 953 y Fw(\))p Fo(E)s Fw(])k(increases)0 1022 y(to)15 b(61.)k(W)l(e)c(see)h(once)f(again)g(a)g(signi\014can)o(t)h (impro)o(v)o(emen)o(t)f(in)h(the)g(case)f(of)g(EM,)f(b)o(y)h(factors)f (of)h(4)g(and)g(6.)71 1090 y(Fig.)27 b(3)18 b(sho)o(ws)f(that)g(the)h (matrix)f Fo(P)24 b Fw(has)18 b(also)g(reduced)h(the)f(largest)f(eigen) o(v)m(alues)j(of)d(the)h(Hessian)h(from)0 1159 y(b)q(et)o(w)o(een)11 b(2000)f(to)g(3000)f(to)h(around)h(1.)18 b(This)11 b(demonstrates)f (clearly)i(the)f(stable)g(con)o(v)o(ergence)g(that)f(is)h(obtained)0 1228 y(via)k(EM,)g(without)g(a)g(line)i(searc)o(h)e(or)f(the)i(need)g (for)e(external)i(selection)h(of)d(a)h(learning)i(stepsize.)71 1297 y(In)e(the)g(remainder)g(of)g(the)g(pap)q(er)g(w)o(e)f(pro)o(vide) i(some)e(theoretical)i(analyses)f(that)f(attempt)g(to)g(shed)h(some)0 1366 y(ligh)o(t)k(on)g(these)g(empirical)h(results.)31 b(T)l(o)18 b(illustrate)i(the)e(issues)i(in)o(v)o(olv)o(ed,)g(consider) f(a)g(degenerate)f(mixture)0 1435 y(problem)f(in)g(whic)o(h)h(the)e (mixture)h(has)f(a)g(single)i(comp)q(onen)o(t.)23 b(\(In)17 b(this)g(case)f Fo(\013)1409 1442 y Fp(1)1443 1435 y Fw(=)f(1.\))23 b(Let)17 b(us)f(furthermore)0 1504 y(assume)g(that)f (the)h(co)o(v)m(ariance)g(matrix)g(is)g(\014xed)h(\(i.e.,)e(only)i(the) f(mean)g(v)o(ector)f Fo(m)h Fw(is)g(to)f(b)q(e)i(estimated\).)22 b(The)0 1573 y(Hessian)e(with)f(resp)q(ect)g(to)g(the)g(mean)g Fo(m)f Fw(is)i Fo(H)i Fw(=)d Fm(\000)p Fo(N)5 b Fw(\006)1035 1556 y Fl(\000)p Fp(1)1101 1573 y Fw(and)19 b(the)g(EM)g(pro)s(jection) f(matrix)h Fo(P)25 b Fw(is)20 b(\006)p Fo(=)m(N)t Fw(.)0 1641 y(F)l(or)g(gradien)o(t)h(ascen)o(t,)h(w)o(e)f(ha)o(v)o(e)f Fo(\024)p Fw([)p Fo(E)688 1625 y Fn(T)715 1641 y Fo(H)t(E)s Fw(])g(=)j Fo(\024)p Fw([\006)957 1625 y Fl(\000)p Fp(1)1004 1641 y Fw(],)f(whic)o(h)f(is)h(larger)f(than)g(one)g(whenev)o(er)g (\006)h Fm(6)p Fw(=)h Fo(cI)t Fw(.)0 1710 y(EM,)e(on)g(the)h(other)f (hand,)i(ac)o(hiev)o(es)f(a)f(condition)i(n)o(um)o(b)q(er)f(of)f(one)g (exactly)h(\()p Fo(\024)p Fw([)p Fo(E)1528 1694 y Fn(T)1555 1710 y Fo(P)6 b(H)t(E)s Fw(])21 b(=)j Fo(\024)p Fw([)p Fo(P)6 b(H)t Fw(])22 b(=)0 1779 y Fo(\024)p Fw([)p Fo(I)t Fw(])17 b(=)g(1)h(and)h Fo(\025)305 1786 y Fn(M)344 1779 y Fw([)p Fo(E)394 1763 y Fn(T)420 1779 y Fo(P)6 b(H)t(E)s Fw(])16 b(=)i(1\).)28 b(Th)o(us,)19 b(EM)e(and)i(Newton's)e(metho)q(d)h (are)g(the)g(same)g(for)g(this)g(simple)0 1848 y(quadratic)12 b(problem.)19 b(F)l(or)10 b(general)i(non-quadratic)g(optimization)g (problems,)h(Newton)e(retains)g(the)h(quadratic)0 1917 y(assumption,)g(yielding)i(fast)c(con)o(v)o(ergence)i(but)g(p)q (ossible)h(div)o(ergence.)20 b(EM)11 b(is)h(a)f(more)g(conserv)m(ativ)o (e)h(algorithm)0 1986 y(that)g(retains)h(the)f(con)o(v)o(ergence)h (guaran)o(tee)f(but)g(also)h(main)o(tains)g(quasi-Newton)g(b)q(eha)o (vior.)19 b(W)l(e)13 b(no)o(w)f(analyze)0 2055 y(this)17 b(b)q(eha)o(vior)g(in)g(more)f(detail.)25 b(W)l(e)17 b(consider)g(the)g(sp)q(ecial)h(case)e(of)g(estimating)h(the)g(means)f (in)i(a)e(Gaussian)0 2124 y(mixture)g(when)f(the)h(Gaussians)f(are)g(w) o(ell)h(separated.)0 2242 y Fd(Theorem)h(2)23 b Fx(Consider)14 b(the)i(EM)f(algorithm)h(in)f(Eq.)20 b(\(3\),)15 b(wher)n(e)g(the)h(p)n (ar)n(ameters)f Fo(\013)1521 2249 y Fn(j)1555 2242 y Fx(and)g Fw(\006)1675 2249 y Fn(j)1709 2242 y Fx(ar)n(e)g(assume)n(d)0 2311 y(to)f(b)n(e)e(known.)20 b(Assume)12 b(that)i(the)g Fo(K)i Fx(Gaussian)d(distributions)g(ar)n(e)g(wel)r(l)g(sep)n(ar)n(ate) n(d,)h(such)f(that)h(for)g(su\016ciently)0 2380 y(lar)n(ge)i Fo(k)h Fx(the)g(p)n(osterior)g(pr)n(ob)n(abilities)e Fo(h)694 2356 y Fp(\()p Fn(k)q Fp(\))694 2393 y Fn(j)743 2380 y Fw(\()p Fo(t)p Fw(\))h Fx(ar)n(e)h(ne)n(arly)e(zer)n(o)h(or)h (one.)k(F)m(or)16 b(such)g Fo(k)q Fx(,)h(the)f(c)n(ondition)g(numb)n (er)0 2449 y(asso)n(ciate)n(d)i(with)h(EM)g(is)f(always)g(smal)r(ler)h (than)g(the)g(c)n(ondition)f(numb)n(er)g(asso)n(ciate)n(d)g(with)h(gr)n (adient)g(asc)n(ent.)p 0 2488 780 2 v 52 2515 a Fj(3)69 2531 y Fi(In)o(terestingly)m(,)13 b(the)f(EM)f(algorithm)j(con)o(v)o (erges)e(so)q(on)g(afterw)o(ard)f(as)h(w)o(ell,)g(sho)o(wing)h(that)e (for)g(this)h(problem)h(EM)e(sp)q(ends)i(little)0 2587 y(time)g(in)h(the)f(region)i(of)d(parameter)i(space)g(in)g(whic)o(h)f (a)g(lo)q(cal)i(analysis)g(is)f(v)n(alid.)952 2817 y Fw(12)p eop %%Page: 13 13 13 12 bop 0 195 a Fx(That)16 b(is:)548 264 y Fo(\024)p Fw([)p Fo(E)624 245 y Fn(T)651 264 y Fo(P)6 b Fw(\(\002)739 245 y Fp(\()p Fn(k)q Fp(\))788 264 y Fw(\))p Fo(H)t Fw(\(\002)901 245 y Fp(\()p Fn(k)q Fp(\))949 264 y Fw(\))p Fo(E)s Fw(])11 b Fo(<)i(\024)p Fw([)p Fo(E)1152 245 y Fn(T)1178 264 y Fo(H)t Fw(\(\002)1273 245 y Fp(\()p Fn(k)q Fp(\))1322 264 y Fw(\))p Fo(E)s Fw(])p Fo(:)0 356 y Fx(F)m(urthermor)n(e,)k Fo(\025)304 363 y Fn(M)343 356 y Fw([)p Fo(E)393 340 y Fn(T)419 356 y Fo(P)6 b Fw(\(\002)507 340 y Fp(\()p Fn(k)q Fp(\))556 356 y Fw(\))p Fo(H)t Fw(\(\002)669 340 y Fp(\()p Fn(k)q Fp(\))717 356 y Fw(\))p Fo(E)s Fw(])15 b Fx(appr)n(o)n(aches)h(one)g(as)g Fo(k)i Fx(go)n(es)d(to)i (in\014nity.)71 525 y Fd(Pro)q(of.)i Fw(The)d(Hessian)g(is)648 669 y Fo(H)g Fw(=)750 535 y Fe(2)750 608 y(6)750 633 y(6)750 658 y(6)750 683 y(6)750 709 y(4)793 563 y Fo(H)831 570 y Fp(11)928 563 y Fo(H)966 570 y Fp(12)1055 563 y Fm(\001)8 b(\001)g(\001)51 b Fo(H)1199 570 y Fp(1)p Fn(K)793 632 y Fo(H)831 639 y Fp(21)928 632 y Fo(H)966 639 y Fp(22)1055 632 y Fm(\001)8 b(\001)g(\001)51 b Fo(H)1199 639 y Fp(2)p Fn(K)824 673 y Fw(.)824 689 y(.)824 706 y(.)959 673 y(.)959 689 y(.)959 706 y(.)1200 673 y(.)1200 689 y(.)1200 706 y(.)785 775 y Fo(H)823 782 y Fn(K)r Fp(1)920 775 y Fo(H)958 782 y Fn(K)r Fp(2)1055 775 y Fm(\001)8 b(\001)g(\001)44 b Fo(H)1192 782 y Fn(K)r(K)1266 535 y Fe(3)1266 608 y(7)1266 633 y(7)1266 658 y(7)1266 683 y(7)1266 709 y(5)1868 669 y Fw(\(17\))0 845 y(where)109 952 y Fo(H)147 959 y Fn(ij)219 952 y Fm(\021)358 921 y Fo(@)385 905 y Fp(2)404 921 y Fo(l)p 301 942 175 2 v 301 985 a(@)s(m)368 992 y Fn(i)381 985 y Fo(@)s(m)448 969 y Fn(T)448 998 y(j)219 1101 y Fw(=)e Fm(\000)p Fw(\(\006)382 1077 y Fp(\()p Fn(k)q Fp(\))382 1114 y Fn(j)430 1101 y Fw(\))448 1082 y Fl(\000)p Fp(1)517 1048 y Fn(N)503 1060 y Fe(X)504 1151 y Fn(t)p Fp(=1)570 1101 y Fo(\016)590 1108 y Fn(ij)621 1101 y Fo(h)647 1077 y Fp(\()p Fn(k)q Fp(\))647 1114 y Fn(j)696 1101 y Fw(\()p Fo(t)p Fw(\))10 b(+)g(\(\006)854 1077 y Fp(\()p Fn(k)q Fp(\))854 1114 y Fn(j)902 1101 y Fw(\))920 1082 y Fl(\000)p Fp(1)967 1101 y Fw([)994 1048 y Fn(N)980 1060 y Fe(X)981 1151 y Fn(t)p Fp(=1)1047 1101 y Fo(\015)1071 1108 y Fn(ij)1101 1101 y Fw(\()p Fo(x)1145 1082 y Fp(\()p Fn(t)p Fp(\))1187 1101 y Fw(\)\()p Fo(x)1249 1082 y Fp(\()p Fn(t)p Fp(\))1301 1101 y Fm(\000)g Fo(m)1386 1108 y Fn(j)1404 1101 y Fw(\)\()p Fo(x)1466 1082 y Fp(\()p Fn(t)p Fp(\))1518 1101 y Fm(\000)h Fo(m)1604 1108 y Fn(i)1618 1101 y Fw(\))1636 1082 y Fn(T)1663 1101 y Fw(]\(\006)1727 1077 y Fp(\()p Fn(k)q Fp(\))1727 1114 y Fn(i)1775 1101 y Fw(\))1793 1082 y Fl(\000)p Fp(1)1868 1101 y Fw(\(18\))0 1246 y(with)16 b Fo(\015)128 1253 y Fn(ij)157 1246 y Fw(\()p Fo(x)201 1229 y Fp(\()p Fn(t)p Fp(\))243 1246 y Fw(\))d(=)g(\()p Fo(\016)360 1253 y Fn(ij)400 1246 y Fm(\000)d Fo(h)471 1222 y Fp(\()p Fn(k)q Fp(\))471 1259 y Fn(i)520 1246 y Fw(\()p Fo(t)p Fw(\)\))p Fo(h)616 1222 y Fp(\()p Fn(k)q Fp(\))616 1259 y Fn(j)665 1246 y Fw(\()p Fo(t)p Fw(\).)19 b(The)d(pro)s(jection)f(matrix)g Fo(P)21 b Fw(is)701 1353 y Fo(P)736 1334 y Fp(\()p Fn(k)q Fp(\))798 1353 y Fw(=)13 b(diag)q([)p Fo(P)978 1329 y Fp(\()p Fn(k)q Fp(\))972 1365 y(11)1026 1353 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(P)1163 1329 y Fp(\()p Fn(k)q Fp(\))1157 1367 y Fn(K)r(K)1223 1353 y Fw(])p Fo(;)0 1460 y Fw(where)743 1544 y Fo(P)778 1520 y Fp(\()p Fn(k)q Fp(\))772 1557 y Fn(j)r(j)839 1544 y Fw(=)13 b(\006)920 1520 y Fp(\()p Fn(k)q Fp(\))920 1557 y Fn(j)969 1544 y Fo(=)1014 1491 y Fn(N)999 1503 y Fe(X)1000 1594 y Fn(t)p Fp(=1)1067 1544 y Fo(h)1093 1520 y Fp(\()p Fn(k)q Fp(\))1093 1557 y Fn(j)1142 1544 y Fw(\()p Fo(t)p Fw(\))p Fo(:)0 1674 y Fw(Giv)o(en)i(that)f Fo(h)255 1650 y Fp(\()p Fn(k)q Fp(\))255 1687 y Fn(j)304 1674 y Fw(\()p Fo(t)p Fw(\)\(1)8 b Fm(\000)h Fo(h)475 1650 y Fp(\()p Fn(k)q Fp(\))475 1687 y Fn(j)524 1674 y Fw(\()p Fo(t)p Fw(\)\))14 b(is)h(negligible)j (for)c(su\016cien)o(tly)h(large)g Fo(k)q Fw(,)f(the)h(second)g(term)f (in)i(Eq.)j(\(18\))14 b(can)0 1743 y(b)q(e)20 b(neglected,)h(yielding)h Fo(H)496 1750 y Fn(ii)541 1743 y Fw(=)d Fm(\000)p Fw(\(\006)681 1719 y Fp(\()p Fn(k)q Fp(\))681 1756 y Fn(j)730 1743 y Fw(\))748 1727 y Fl(\000)p Fp(1)802 1711 y Fe(P)846 1724 y Fn(N)846 1754 y(t)p Fp(=1)914 1743 y Fo(h)940 1719 y Fp(\()p Fn(k)q Fp(\))940 1756 y Fn(j)989 1743 y Fw(\()p Fo(t)p Fw(\))g(and)g Fo(H)k Fw(=)d(diag)q([)p Fo(H)1403 1750 y Fp(11)1439 1743 y Fo(;)8 b Fm(\001)g(\001)g(\001)d Fo(;)j(H)1579 1750 y Fn(K)r(K)1644 1743 y Fw(].)32 b(This)20 b(implies)0 1812 y(that)14 b Fo(P)6 b(H)17 b Fw(=)c Fm(\000)p Fo(I)t Fw(,)i(and)g(th)o(us)g Fo(\024)p Fw([)p Fo(P)6 b(H)t Fw(])12 b(=)h(1,)i(whereas)g Fo(\024)p Fw([)p Fo(H)t Fw(])c Fm(6)p Fw(=)i(1.)803 b Fc(2)71 1950 y Fw(This)11 b(theorem,)f(although)h(restrictiv)o(e)g(in)g(its)g(assumptions,)g(giv) o(es)f(some)g(indication)j(as)d(to)f(wh)o(y)i(the)f(pro)s(jec-)0 2019 y(tion)k(matrix)g(in)g(the)g(EM)g(algorithm)g(app)q(ears)g(to)f (condition)i(the)f(Hessian,)h(yielding)h(impro)o(v)o(ed)e(con)o(v)o (ergence.)0 2087 y(In)h(fact,)e(w)o(e)h(conjecture)h(that)f(the)g (theorem)g(can)g(b)q(e)h(extended)h(to)d(apply)i(more)f(widely)l(,)i (in)f(particular)g(to)f(the)0 2156 y(case)e(of)f(the)h(full)i(EM)d(up)q (date)h(in)h(whic)o(h)g(the)f(mixing)h(prop)q(ortions)f(and)g(co)o(v)m (ariances)g(are)g(estimated,)g(and)g(also,)0 2225 y(within)j(limits,)f (to)f(cases)g(in)i(whic)o(h)f(the)f(means)h(are)f(not)g(w)o(ell)h (separated.)19 b(T)l(o)13 b(obtain)h(an)f(initial)j(indication)f(as)0 2294 y(to)g(p)q(ossible)i(conditions)g(that)e(can)h(b)q(e)g(usefully)i (imp)q(osed)e(on)g(the)g(separation)f(of)g(the)h(mixture)g(comp)q(onen) o(ts,)0 2363 y(w)o(e)j(ha)o(v)o(e)f(studied)i(the)f(case)g(in)h(whic)o (h)f(the)g(second)h(term)e(in)i(Eq.)31 b(\(18\))17 b(is)j(neglected)g (only)f(for)f Fo(H)1781 2370 y Fn(ii)1826 2363 y Fw(and)h(is)0 2432 y(retained)h(for)g(the)f Fo(H)376 2439 y Fn(ij)426 2432 y Fw(comp)q(onen)o(ts,)i(where)f Fo(j)i Fm(6)p Fw(=)f Fo(i)p Fw(.)33 b(Consider,)21 b(for)e(example,)i(the)f(case)g(of)f(a)g (univ)m(ariate)0 2501 y(mixture)d(ha)o(ving)g(t)o(w)o(o)f(mixture)h (comp)q(onen)o(ts.)22 b(F)l(or)15 b(\014xed)i(mixing)g(prop)q(ortions)f (and)g(\014xed)g(co)o(v)m(ariances,)h(the)0 2570 y(Hessian)f(matrix)f (\(Eq.)k(17\))14 b(b)q(ecomes:)795 2659 y Fo(H)i Fw(=)897 2587 y Fe(")929 2624 y Fo(h)955 2631 y Fp(11)1038 2624 y Fo(h)1064 2631 y Fp(12)929 2693 y Fo(h)955 2700 y Fp(21)1038 2693 y Fo(h)1064 2700 y Fp(22)1109 2587 y Fe(#)1141 2659 y Fo(;)952 2817 y Fw(13)p eop %%Page: 14 14 14 13 bop 0 195 a Fw(and)15 b(the)h(pro)s(jection)f(matrix)g(\(Eq.)k (18\))14 b(b)q(ecomes:)753 330 y Fo(P)19 b Fw(=)849 258 y Fe(")881 295 y Fm(\000)p Fo(h)942 276 y Fl(\000)p Fp(1)942 308 y(11)1078 295 y Fw(0)924 364 y(0)88 b Fm(\000)p Fo(h)1096 345 y Fl(\000)p Fp(1)1096 377 y(22)1151 258 y Fe(#)1183 330 y Fo(;)0 462 y Fw(where)659 546 y Fo(h)685 553 y Fn(ii)724 546 y Fw(=)13 b Fm(\000)848 515 y Fw(1)p 812 536 95 2 v 812 589 a Fo(\033)840 565 y Fp(2\()p Fn(k)q Fp(\))838 602 y Fn(i)933 493 y(N)919 506 y Fe(X)920 596 y Fn(t)p Fp(=1)986 546 y Fo(h)1012 522 y Fp(\()p Fn(k)q Fp(\))1012 559 y Fn(i)1061 546 y Fw(\()p Fo(t)p Fw(\))p Fo(;)22 b(i)12 b Fw(=)h(1)p Fo(;)8 b Fw(2)0 665 y(and)241 749 y Fo(h)267 756 y Fn(ij)310 749 y Fw(=)446 718 y(1)p 363 738 189 2 v 363 792 a Fo(\033)391 768 y Fp(2\()p Fn(k)q Fp(\))389 805 y Fn(i)457 792 y Fo(\033)485 768 y Fp(2\()p Fn(k)q Fp(\))483 805 y Fn(j)578 696 y(N)564 708 y Fe(X)565 799 y Fn(t)p Fp(=1)624 749 y Fw(\(1)h Fm(\000)i Fo(h)746 725 y Fp(\()p Fn(k)q Fp(\))746 762 y Fn(i)794 749 y Fw(\()p Fo(t)p Fw(\)\))p Fo(h)890 725 y Fp(\()p Fn(k)q Fp(\))890 762 y Fn(j)939 749 y Fw(\()p Fo(t)p Fw(\)\()p Fo(x)1035 730 y Fp(\()p Fn(t)p Fp(\))1087 749 y Fm(\000)f Fo(m)1172 756 y Fn(j)1190 749 y Fw(\))1208 730 y Fn(T)1236 749 y Fw(\()p Fo(x)1280 730 y Fp(\()p Fn(t)p Fp(\))1332 749 y Fm(\000)g Fo(m)1417 756 y Fn(i)1431 749 y Fw(\))p Fo(;)22 b(i)12 b Fm(6)p Fw(=)h Fo(j)i Fw(=)e(1)p Fo(;)8 b Fw(2)0 877 y(If)20 b Fo(H)j Fw(is)d(negativ)o(e)g(de\014nite,) i(\(i.e.,)e Fo(h)660 884 y Fp(11)698 877 y Fo(h)724 884 y Fp(22)774 877 y Fm(\000)14 b Fo(h)849 884 y Fp(12)886 877 y Fo(h)912 884 y Fp(21)970 877 y Fo(<)21 b Fw(0\),)f(then)g(w)o(e)f (can)h(sho)o(w)f(that)g(the)h(conclusions)h(of)0 946 y(Theorem)c(2)g(remain)h(true,)f(ev)o(en)g(for)g(Gaussians)g(that)g (are)g(not)f(necessarily)j(w)o(ell-separated.)27 b(The)17 b(pro)q(of)g(is)0 1015 y(ac)o(hiev)o(ed)f(via)g(the)f(follo)o(wing)h (lemma:)0 1133 y Fd(Lemma)h(1)23 b Fx(Consider)16 b(the)g(p)n(ositive)g (de\014nite)f(matrix)809 1266 y Fw(\006)e(=)903 1194 y Fe(")935 1231 y Fo(\033)961 1238 y Fp(11)1044 1231 y Fo(\033)1070 1238 y Fp(12)935 1300 y Fo(\033)961 1307 y Fp(21)1044 1300 y Fo(\033)1070 1307 y Fp(22)1115 1194 y Fe(#)0 1407 y Fx(F)m(or)j(the)g(diagonal)h(matrix)g Fo(B)e Fw(=)e(diag)q([)p Fo(\033)710 1388 y Fl(\000)p Fp(1)708 1419 y(11)756 1407 y Fo(;)8 b(\033)805 1388 y Fl(\000)p Fp(1)803 1419 y(22)851 1407 y Fw(])p Fx(,)16 b(we)g(have)h Fo(\024)p Fw([)p Fo(B)r Fw(\006])c Fo(<)g(\024)p Fw([\006])p Fx(.)71 1585 y Fd(Pro)q(of.)19 b Fw(The)d(eigen)o(v)m (alues)h(of)e(\006)g(are)f(the)i(ro)q(ots)e(of)h(\()p Fo(\033)1026 1592 y Fp(11)1073 1585 y Fm(\000)10 b Fo(\025)p Fw(\)\()p Fo(\033)1207 1592 y Fp(22)1254 1585 y Fm(\000)g Fo(\025)p Fw(\))f Fm(\000)i Fo(\033)1425 1592 y Fp(21)1462 1585 y Fo(\033)1488 1592 y Fp(12)1538 1585 y Fw(=)i(0,)i(whic)o(h)h (giv)o(es)532 1699 y Fo(\025)559 1706 y Fn(M)639 1699 y Fw(=)721 1668 y Fo(\033)747 1675 y Fp(11)794 1668 y Fw(+)11 b Fo(\033)866 1675 y Fp(22)913 1668 y Fw(+)g Fo(\015)p 721 1689 264 2 v 841 1730 a Fw(2)538 1804 y Fo(\025)565 1811 y Fn(m)639 1804 y Fw(=)721 1773 y Fo(\033)747 1780 y Fp(11)794 1773 y Fw(+)g Fo(\033)866 1780 y Fp(22)913 1773 y Fm(\000)g Fo(\015)p 721 1794 V 841 1835 a Fw(2)572 1907 y Fo(\015)43 b Fw(=)716 1855 y Fe(q)p 757 1855 643 2 v 757 1907 a Fw(\()p Fo(\033)801 1914 y Fp(11)849 1907 y Fw(+)10 b Fo(\033)920 1914 y Fp(22)957 1907 y Fw(\))975 1893 y Fp(2)1005 1907 y Fm(\000)g Fw(4\()p Fo(\033)1117 1914 y Fp(11)1154 1907 y Fo(\033)1180 1914 y Fp(22)1227 1907 y Fm(\000)h Fo(\033)1299 1914 y Fp(21)1336 1907 y Fo(\033)1362 1914 y Fp(12)1400 1907 y Fw(\))0 2021 y(and)736 2136 y Fo(\024)p Fw([\006])41 b(=)944 2105 y Fo(\033)970 2112 y Fp(11)1017 2105 y Fw(+)11 b Fo(\033)1089 2112 y Fp(22)1136 2105 y Fw(+)g Fo(\015)p 944 2125 264 2 v 944 2167 a(\033)970 2174 y Fp(11)1017 2167 y Fw(+)g Fo(\033)1089 2174 y Fp(22)1136 2167 y Fm(\000)g Fo(\015)0 2260 y Fw(The)16 b(condition)h(n)o(um)o(b)q(er)f Fo(\024)p Fw([\006])f(can)g(b)q(e)i(written)e(as)h Fo(\024)p Fw([\006])c(=)i(\(1) c(+)h Fo(s)p Fw(\))p Fo(=)p Fw(\(1)e Fm(\000)i Fo(s)p Fw(\))i Fm(\021)h Fo(f)5 b Fw(\()p Fo(s)p Fw(\),)15 b(where)h Fo(s)g Fw(is)g(de\014ned)h(as)0 2328 y(follo)o(ws:)679 2412 y Fo(s)c Fw(=)761 2336 y Fe(s)p 802 2336 456 2 v 802 2412 a Fw(1)d Fm(\000)885 2381 y Fw(4\()p Fo(\033)952 2388 y Fp(11)989 2381 y Fo(\033)1015 2388 y Fp(22)1063 2381 y Fm(\000)g Fo(\033)1134 2388 y Fp(21)1171 2381 y Fo(\033)1197 2388 y Fp(12)1235 2381 y Fw(\))p 885 2401 367 2 v 950 2443 a(\()p Fo(\033)994 2450 y Fp(11)1041 2443 y Fw(+)h Fo(\033)1113 2450 y Fp(22)1150 2443 y Fw(\))1168 2430 y Fp(2)1257 2412 y Fo(:)71 2524 y Fw(F)l(urthermore,)18 b(the)g(eigen)o(v)m(alues)i(of)e Fo(B)r Fw(\006)h(are)f(the)h(ro)q(ots) e(of)h(\(1)12 b Fm(\000)g Fo(\025)p Fw(\)\(1)f Fm(\000)i Fo(\025)p Fw(\))e Fm(\000)i Fw(\()p Fo(\033)1540 2531 y Fp(21)1577 2524 y Fo(\033)1603 2531 y Fp(12)1640 2524 y Fw(\))p Fo(=)p Fw(\()p Fo(\033)1725 2531 y Fp(11)1761 2524 y Fo(\033)1787 2531 y Fp(22)1825 2524 y Fw(\))k(=)h(0,)0 2593 y(whic)o(h)k(giv)o(es)g Fo(\025)281 2600 y Fn(M)342 2593 y Fw(=)h(1)14 b(+)487 2556 y Fe(p)p 528 2556 330 2 v 528 2593 a Fw(\()p Fo(\033)572 2600 y Fp(21)609 2593 y Fo(\033)635 2600 y Fp(12)673 2593 y Fw(\))p Fo(=)p Fw(\()p Fo(\033)758 2600 y Fp(11)794 2593 y Fo(\033)820 2600 y Fp(22)857 2593 y Fw(\))21 b(and)h Fo(\025)1018 2600 y Fn(m)1073 2593 y Fw(=)h(1)14 b Fm(\000)1218 2556 y Fe(p)p 1259 2556 347 2 v 1259 2593 a Fw(\()p Fo(\033)1303 2600 y Fp(21)1340 2593 y Fo(\033)1366 2600 y Fp(12)1403 2593 y Fw(\))p Fo(=)p Fw(\()p Fo(\033)1488 2600 y Fp(11)1525 2593 y Fo(\033)1551 2600 y Fp(22)1588 2593 y Fw(\).)38 b(Th)o(us,)22 b(de\014ning)0 2662 y Fo(r)13 b Fw(=)82 2625 y Fe(p)p 124 2625 V 37 x Fw(\()p Fo(\033)168 2669 y Fp(21)205 2662 y Fo(\033)231 2669 y Fp(12)268 2662 y Fw(\))p Fo(=)p Fw(\()p Fo(\033)353 2669 y Fp(11)390 2662 y Fo(\033)416 2669 y Fp(22)453 2662 y Fw(\),)h(w)o(e)h(ha)o(v)o(e) g Fo(\024)p Fw([)p Fo(B)r Fw(\006])e(=)g(\(1)c(+)i Fo(r)q Fw(\))p Fo(=)p Fw(\(1)d Fm(\000)j Fo(r)q Fw(\))h(=)h Fo(f)5 b Fw(\()p Fo(r)q Fw(\).)952 2817 y(14)p eop %%Page: 15 15 15 14 bop 71 195 a Fw(W)l(e)15 b(no)o(w)g(examine)h(the)f(quotien)o(t)g Fo(s=r)q Fw(:)641 282 y Fo(s)p 641 302 22 2 v 641 344 a(r)680 312 y Fw(=)733 282 y(1)p 733 302 23 2 v 733 344 a Fo(r)761 236 y Fe(s)p 803 236 511 2 v 76 x Fw(1)9 b Fm(\000)1008 282 y Fw(4\(1)g Fm(\000)i Fo(r)1149 268 y Fp(2)1168 282 y Fw(\))p 886 302 423 2 v 886 344 a(\()p Fo(\033)930 351 y Fp(11)977 344 y Fw(+)f Fo(\033)1048 351 y Fp(22)1086 344 y Fw(\))1104 330 y Fp(2)1123 344 y Fo(=)p Fw(\()p Fo(\033)1190 351 y Fp(11)1227 344 y Fo(\033)1253 351 y Fp(22)1290 344 y Fw(\))0 426 y(Giv)o(en)19 b(that)f(\()p Fo(\033)281 433 y Fp(11)330 426 y Fw(+)13 b Fo(\033)404 433 y Fp(22)441 426 y Fw(\))459 409 y Fp(2)479 426 y Fo(=)p Fw(\()p Fo(\033)546 433 y Fp(11)582 426 y Fo(\033)608 433 y Fp(22)646 426 y Fw(\))18 b Fm(\025)g Fw(4,)h(w)o(e)f(ha)o(v)o(e)974 408 y Fn(s)p 974 415 17 2 v 974 441 a(r)1014 426 y Fo(>)1072 408 y Fp(1)p 1072 415 18 2 v 1072 441 a Fn(r)1095 389 y Fe(p)p 1137 389 234 2 v 37 x Fw(1)9 b Fm(\000)i Fw(\(1)e Fm(\000)i Fo(r)1333 413 y Fp(2)1352 426 y Fw(\))18 b(=)h(1.)30 b(That)18 b(is,)h Fo(s)g(>)f(r)q Fw(.)30 b(Since)0 495 y Fo(f)5 b Fw(\()p Fo(x)p Fw(\))19 b(=)h(\(1)12 b(+)h Fo(x)p Fw(\))p Fo(=)p Fw(\(1)f Fm(\000)h Fo(x)p Fw(\))19 b(is)h(a)f(monotonically)h (increasing)h(function)f(for)f Fo(x)g(>)h Fw(0,)g(w)o(e)f(ha)o(v)o(e)g Fo(f)5 b Fw(\()p Fo(s)p Fw(\))19 b Fo(>)h(f)5 b Fw(\()p Fo(r)q Fw(\).)0 563 y(Therefore,)15 b Fo(\024)p Fw([)p Fo(B)r Fw(\006])d Fo(<)h(\024)p Fw([\006].)1417 b Fc(2)71 701 y Fw(W)l(e)23 b(think)g(that)g(it)g(should)h(b)q(e)f(p)q(ossible)i (to)d(generalize)j(this)e(lemma)g(b)q(ey)o(ond)h(the)f(univ)m(ariate,)j (t)o(w)o(o-)0 770 y(comp)q(onen)o(t)15 b(case,)f(thereb)o(y)h(w)o(eak)o (ening)g(the)f(conditions)i(on)f(separabilit)o(y)g(in)h(Theorem)e(2)g (in)i(a)e(more)g(general)0 839 y(setting.)0 995 y Fq(6)69 b(Conclusions)0 1109 y Fw(In)17 b(this)g(pap)q(er)g(w)o(e)f(ha)o(v)o(e) g(pro)o(vided)i(a)e(comparativ)o(e)g(analysis)h(of)f(algorithms)h(for)e (the)i(learning)h(of)e(Gaussian)0 1178 y(mixtures.)k(W)l(e)14 b(ha)o(v)o(e)g(fo)q(cused)i(on)e(the)g(EM)g(algorithm)h(and)f(ha)o(v)o (e)h(forged)e(a)i(link)g(b)q(et)o(w)o(een)g(EM)f(and)h(gradien)o(t)0 1247 y(metho)q(ds)i(via)g(the)h(pro)s(jection)e(matrix)h Fo(P)6 b Fw(.)26 b(W)l(e)17 b(ha)o(v)o(e)g(also)g(analyzed)g(the)h(con) o(v)o(ergence)f(of)f(EM)h(in)h(terms)e(of)0 1315 y(prop)q(erties)g(of)f (the)g(matrix)g Fo(P)21 b Fw(and)16 b(the)f(e\013ect)g(that)g Fo(P)21 b Fw(has)15 b(on)g(the)h(lik)o(eliho)q(o)q(d)i(surface.)71 1384 y(EM)10 b(has)h(a)g(n)o(um)o(b)q(er)g(of)f(prop)q(erties)i(that)e (mak)o(e)h(it)g(a)g(particularly)h(attractiv)o(e)e(algorithm)h(for)f (mixture)i(mo)q(d-)0 1453 y(els.)19 b(It)12 b(enjo)o(ys)f(automatic)g (satisfaction)g(of)g(probabilistic)j(constrain)o(ts,)d(monotonic)h(con) o(v)o(ergence)f(without)h(the)0 1522 y(need)17 b(to)e(set)h(a)f (learning)i(rate,)e(and)h(lo)o(w)g(computational)g(o)o(v)o(erhead.)22 b(Although)16 b(EM)f(has)h(the)g(reputation)g(of)0 1591 y(b)q(eing)e(a)f(slo)o(w)g(algorithm,)h(w)o(e)e(feel)j(that)d(in)i(the) f(mixture)h(setting)f(the)g(slo)o(wness)h(of)e(EM)h(has)g(b)q(een)i(o)o (v)o(erstated.)0 1660 y(Although)i(EM)g(can)g(indeed)h(con)o(v)o(erge)f (slo)o(wly)g(for)f(problems)h(in)h(whic)o(h)g(the)f(mixture)g(comp)q (onen)o(ts)g(are)f(not)0 1729 y(w)o(ell)g(separated,)e(the)i(Hessian)f (is)h(p)q(o)q(orly)f(conditioned)i(for)e(suc)o(h)g(problems)h(and)f(th) o(us)g(other)f(gradien)o(t-based)0 1798 y(algorithms)h(\(including)j (Newton's)d(metho)q(d\))h(are)f(also)g(lik)o(ely)j(to)d(p)q(erform)g(p) q(o)q(orly)l(.)21 b(Moreo)o(v)o(er,)14 b(if)i(one's)f(con-)0 1866 y(cern)k(is)g(con)o(v)o(ergence)f(in)h(lik)o(eliho)q(o)q(d,)j (then)c(EM)g(generally)i(p)q(erforms)e(w)o(ell)h(ev)o(en)g(for)e(these) i(ill-conditione)q(d)0 1935 y(problems.)h(Indeed)15 b(the)f(algorithm)g (pro)o(vides)g(a)g(certain)g(amoun)o(t)f(of)h(safet)o(y)f(in)h(suc)o(h) g(cases,)g(despite)h(the)f(p)q(o)q(or)0 2004 y(conditioning.)33 b(It)19 b(is)g(also)g(imp)q(ortan)o(t)f(to)g(emphasize)i(that)e(the)h (case)g(of)g(p)q(o)q(orly)g(separated)g(mixture)g(com-)0 2073 y(p)q(onen)o(ts)e(can)f(b)q(e)h(view)o(ed)g(as)f(a)g(problem)h(in) g(mo)q(del)g(selection)h(\(to)q(o)e(man)o(y)f(mixture)i(comp)q(onen)o (ts)f(are)g(b)q(eing)0 2142 y(included)i(in)e(the)f(mo)q(del\),)h(and)f (should)h(b)q(e)g(handled)h(b)o(y)e(regularization)h(tec)o(hniques.)71 2211 y(The)11 b(fact)g(that)f(EM)h(is)h(a)f(\014rst)g(order)f (algorithm)i(certainly)g(implies)h(that)e(EM)g(is)g(no)h(panacea,)f (but)h(do)q(es)f(not)0 2280 y(imply)h(that)e(EM)h(has)g(no)g(adv)m(an)o (tages)f(o)o(v)o(er)g(gradien)o(t)h(ascen)o(t)g(or)f(sup)q(erlinear)j (metho)q(ds.)18 b(First,)12 b(it)f(is)g(imp)q(ortan)o(t)0 2349 y(to)k(appreciate)h(that)f(con)o(v)o(ergence)h(rate)f(results)g (are)h(generally)g(obtained)h(for)e(unconstrained)h(optimization,)0 2417 y(and)11 b(are)g(not)g(necessarily)i(indicativ)o(e)g(of)e(p)q (erformance)g(on)g(constrained)h(optimization)g(problems.)19 b(Also,)12 b(as)f(w)o(e)0 2486 y(ha)o(v)o(e)i(demonstrated,)g(there)h (are)f(conditions)h(under)g(whic)o(h)h(the)e(condition)i(n)o(um)o(b)q (er)f(of)f(the)g(e\013ectiv)o(e)h(Hessian)0 2555 y(of)i(the)g(EM)g (algorithm)g(tends)g(to)o(w)o(ard)f(one,)h(sho)o(wing)h(that)e(EM)h (can)g(appro)o(ximate)g(a)g(sup)q(erlinear)i(metho)q(d.)0 2624 y(Finally)l(,)c(in)f(cases)f(of)f(a)h(p)q(o)q(orly)g(conditioned)i (Hessian,)f(sup)q(erlinear)h(con)o(v)o(ergence)e(is)g(not)g (necessarily)h(a)f(virtue.)0 2693 y(In)k(suc)o(h)f(cases)g(man)o(y)g (optimization)h(sc)o(hemes,)f(includin)q(g)i(EM,)e(essen)o(tially)i (rev)o(ert)d(to)h(gradien)o(t)g(ascen)o(t.)952 2817 y(15)p eop %%Page: 16 16 16 15 bop 71 195 a Fw(W)l(e)13 b(feel)h(that)f(EM)f(will)j(con)o(tin)o (ue)f(to)f(pla)o(y)g(an)g(imp)q(ortan)o(t)g(role)h(in)g(the)f(dev)o (elopmen)o(t)h(of)f(learning)h(systems)0 264 y(that)k(emphasize)i(the)g (predictiv)o(e)g(asp)q(ect)f(of)g(data)f(mo)q(deling.)33 b(EM)18 b(has)h(indeed)i(pla)o(y)o(ed)f(a)e(critical)j(role)e(in)0 333 y(the)i(dev)o(elopmen)o(t)h(of)e(hidden)j(Mark)o(o)o(v)c(mo)q(dels) j(\(HMM's\),)e(an)h(imp)q(ortan)o(t)g(example)g(of)g(predictiv)o(e)h (data)0 402 y(mo)q(deling.)192 385 y Fp(4)243 402 y Fw(EM)c(generally)i (con)o(v)o(erges)e(rapidly)h(in)h(this)f(setting.)29 b(Similarly)l(,)22 b(in)d(the)f(case)h(of)f(hierarc)o(hical)0 471 y(mixtures)g(of)f(exp)q(erts)h(the)g(empirical)i(results)e(on)f (con)o(v)o(ergence)h(in)h(lik)o(eliho)q(o)q(d)h(ha)o(v)o(e)e(b)q(een)h (quite)f(promising)0 539 y(\(Jordan)j(&)h(Jacobs,)h(1994;)g(W)l (aterhouse)e(&)g(Robinson,)j(1994\).)37 b(Finally)l(,)25 b(EM)c(can)g(pla)o(y)h(an)f(imp)q(ortan)o(t)0 608 y(conceptual)16 b(role)e(as)h(an)f(organizing)h(principle)i(in)f(the)e(design)i(of)e (learning)i(algorithms.)j(Its)c(role)f(in)i(this)f(case)0 677 y(is)i(to)e(fo)q(cus)h(atten)o(tion)g(on)g(the)g(\\missing)h(v)m (ariables")g(in)g(the)f(problem.)23 b(This)17 b(clari\014es)g(the)f (structure)g(of)g(the)0 746 y(algorithm)e(and)h(in)o(vites)g (comparisons)f(with)h(statistical)f(ph)o(ysics,)h(where)g(missing)g(v)m (ariables)g(often)f(pro)o(vide)h(a)0 815 y(p)q(o)o(w)o(erful)g (analytic)h(to)q(ol.)0 949 y Fb(Ac)n(kno)n(wledgemen)n(t)0 1048 y Fs(This)d(pro)r(ject)i(w)o(as)e(supp)q(orted)i(in)e(part)g(b)o (y)g(the)h(HK)g(R)o(GC)e(Earmark)o(ed)h(Gran)o(t)g(CUHK250/94E,)f(b)o (y)h(a)g(gran)o(t)h(from)d(the)0 1117 y(McDonnell-P)o(ew)16 b(F)m(oundation,)e(b)o(y)i(a)f(gran)o(t)h(from)e(A)m(TR)h(Human)f (Information)f(Pro)q(cessing)k(Researc)o(h)g(Lab)q(oratories,)0 1185 y(b)o(y)d(a)g(gran)o(t)f(from)g(Siemens)g(Corp)q(oration,)g(b)o(y) h(gran)o(t)g(IRI-9013991)e(from)g(the)j(National)d(Science)k(F)m (oundation,)c(and)i(b)o(y)0 1254 y(gran)o(t)f(N00014-90-J-1942)d(from)i (the)h(O\016ce)h(of)f(Na)o(v)n(al)e(Researc)o(h.)19 b(Mic)o(hael)13 b(I.)g(Jordan)g(is)g(an)g(NSF)g(Presiden)o(tial)h(Y)m(oung)0 1323 y(In)o(v)o(estigator.)0 1451 y Fd(References)0 1536 y Fs(Amari,)g(S.)g(\(in)i(press\))h(Information)12 b(geometry)j(of)g (the)h(EM)g(and)f(em)f(algorithms)f(for)i(neural)h(net)o(w)o(orks,)g Fa(Neur)n(al)f(Net-)0 1596 y(works)p Fs(.)0 1681 y(Baum,)f(L.E.,)h(and) g(Sell,)g(G.R.)f(\(1968\),)h(Gro)o(wth)g(transformation)f(for)h (functions)g(on)h(manifolds,)c Fa(Pac.)24 b(J.)16 b(Math.,)h(27)p Fs(,)0 1741 y(211-227.)0 1832 y(Bengio,)h(Y.,)g(and)g(F)m(rasconi,)g(P) m(.,)g(\(1995\),)f(An)h(input-output)g(HMM)g(arc)o(hitecture.)31 b Fa(A)n(dvanc)n(es)20 b(in)e(Neur)n(al)g(Informa-)0 1893 y(tion)f(Pr)n(o)n(c)n(essing)f(Systems)h(6)p Fs(,)e(eds.,)h(T)m (esauro,)g(G.,)f(T)m(ouretzky)m(,)g(D.S.,)g(and)g(Alsp)q(ector,)i(J.,)f (San)f(Mateo,)h(CA:)f(Morgan)0 1953 y(Kaufmann.)0 2044 y(Bo)o(yles,)i(R.A.)e(\(1983\),)h(On)h(the)g(con)o(v)o(ergence)h(of)e (the)h(EM)f(algorithm,)e Fa(J.)j(of)g(R)n(oyal)h(Statistic)n(al)e(So)n (ciety,)i(B45)p Fs(,)f(No.1,)0 2104 y(47-50.)0 2189 y(Dempster,)f(A.P)m (.,)f(Laird,)h(N.M.,)f(and)h(Rubin,)f(D.B.)g(\(1977\),)h(Maxim)o(um)c (lik)o(eliho)q(o)q(d)i(from)g(incomplete)h(data)h(via)f(the)0 2250 y(EM)f(algorithm,)d Fa(J.)j(of)h(R)n(oyal)g(Statistic)n(al)f(So)n (ciety,)h(B39)p Fs(,)f(1-38.)0 2334 y(Ghahramani,)d(Z,)j(and)g(Jordan,) f(M.I.)g(\(1994\),)g(F)m(unction)h(appro)o(ximation)d(via)i(densit)o(y) h(estimation)f(using)g(the)i(EM)f(ap-)0 2395 y(proac)o(h,)c Fa(A)n(dvanc)n(es)i(in)f(Neur)n(al)f(Information)h(Pr)n(o)n(c)n(essing) g(Systems)g(6)p Fs(,)f(eds.,)h(Co)o(w)o(an,)e(J.D.,)g(T)m(esauro,)h (G.,)f(and)h(Alsp)q(ector,)0 2455 y(J.,)j(San)h(Mateo,)f(CA:)h(Morgan)f (Kaufmann,)f(120-127.)p 0 2494 780 2 v 52 2521 a Fj(4)69 2537 y Fi(In)g(most)g(applications)j(of)c(HMM's,)g(the)h(\\parameter)h (estimation")h(pro)q(cess)e(is)g(emplo)o(y)o(ed)i(solely)f(to)f(yield)h (mo)q(dels)g(with)f(high)0 2593 y(lik)o(eliho)r(o)q(d;)k(the)d (parameters)h(are)f(not)g(generally)i(endo)o(w)o(ed)f(with)f(a)g (particular)j(meaning.)952 2817 y Fw(16)p eop %%Page: 17 17 17 16 bop 0 195 a Fs(Jordan,)20 b(M.I.)f(and)g(Jacobs,)h(R.A.)e (\(1994\),)i(Hierarc)o(hical)f(mixtures)f(of)h(exp)q(erts)i(and)e(the)h (EM)f(algorithm.)31 b Fa(Neur)n(al)0 256 y(Computation)15 b(6)p Fs(,)f(181-214.)0 346 y(Jordan,)h(M.I.)f(and)g(Xu,)h(L.)f(\(in)h (press\),)h(Con)o(v)o(ergence)g(results)g(for)e(the)i(EM)f(approac)o(h) f(to)h(mixtures-of-exp)q(erts)g(arc)o(hi-)0 407 y(tectures,)h Fa(Neur)n(al)e(Networks)p Fs(.)0 497 y(Levinson,)h(S.E.,)g(Rabiner,)g (L.R.,)f(and)i(Sondhi,)e(M.M.)h(\(1983\),)g(An)g(in)o(tro)q(duction)h (to)f(the)h(application)e(of)h(the)h(theory)0 558 y(of)h(probabilistic) h(functions)g(of)f(Mark)o(o)o(v)h(pro)q(cess)i(to)d(automatic)g(sp)q (eec)o(h)j(recognition,)e Fa(The)h(Bel)r(l)f(System)h(T)m(e)n(chnic)n (al)0 619 y(Journal,)c(62)p Fs(,)e(1035-1072.)0 709 y(Neal,)g(R.)g(N.)h (and)g(Hin)o(ton,)e(G.)h(E.)h(\(1993\),)f Fa(A)i(new)f(view)h(of)g(the) g(EM)g(algorithm)f(that)h(justi\014es)g(incr)n(emental)f(and)i(other)0 770 y(variants)p Fs(,)d(Univ)o(ersit)o(y)h(of)f(T)m(oron)o(to,)g (Departmen)o(t)g(of)g(Computer)h(Science)h(preprin)o(t.)0 860 y(No)o(wlan,)g(S.J.)g(\(1991\).)22 b Fa(Soft)17 b(c)n(omp)n (etitive)f(adaptation:)23 b(Neur)n(al)16 b(network)g(le)n(arning)g (algorithms)g(b)n(ase)n(d)g(on)h(\014tting)g(sta-)0 921 y(tistic)n(al)d(mixtur)n(es)p Fs(.)j(T)m(ec)o(h.)h(Rep.)g (CMU-CS-91-126,)12 b(CMU,)i(Pittsburgh,)g(P)m(A.)0 1011 y(Redner,)20 b(R.A.,)d(and)h(W)m(alk)o(er,)f(H.F.)h(\(1984\),)g (Mixture)g(densities,)i(maxim)n(um)14 b(lik)o(eliho)q(o)q(d,)j(and)h (the)h(EM)f(algorithm,)0 1072 y Fa(SIAM)d(R)n(eview)g(26)p Fs(,)f(195-239.)0 1186 y(Titterington,)k(D.M.)e(\(1984\),)i(Recursiv)o (e)g(parameter)f(estimation)f(using)i(incomplete)e(data,)i Fa(J.)g(of)g(R)n(oyal)h(Statistic)n(al)0 1247 y(So)n(ciety,)c(B46)p Fs(,)f(257-267.)0 1361 y(T)m(resp,)k(V,)e(Ahmad,)f(S.)h(and)h (Neuneier,)h(R.)e(\(1994\),)g(T)m(raining)f(neural)i(net)o(w)o(orks)g (with)f(de\014cien)o(t)i(data,)f Fa(A)n(dvanc)n(es)h(in)0 1421 y(Neur)n(al)f(Information)h(Pr)n(o)n(c)n(essing)g(Systems)g(6)p Fs(,)f(eds.,)h(Co)o(w)o(an,)e(J.D.,)h(T)m(esauro,)g(G.,)g(and)f(Alsp)q (ector,)j(J.,)e(San)g(Mateo,)0 1482 y(CA:)c(Morgan)h(Kaufmann,)e (128-135.)0 1572 y(W)m(aterhouse,)19 b(S.)f(R.,)g(and)g(Robinson,)g(A.) g(J.,)h(\(1994\),)f(Classi\014cation)f(using)h(hierarc)o(hical)g (mixtures)g(of)g(exp)q(erts,)i(in)0 1633 y Fa(IEEE)15 b(Workshop)h(on)f(Neur)n(al)g(Networks)f(for)g(Signal)h(Pr)n(o)n(c)n (essing)p Fs(.)0 1724 y(W)m(u.)32 b(C.F.)18 b(J.)g(\(1983\),)h(On)g (the)h(con)o(v)o(ergence)g(prop)q(erties)g(of)f(the)g(EM)g(algorithm,)e Fa(The)i(A)o(nnals)h(of)f(Statistics,)h(11)p Fs(,)0 1784 y(95-103.)0 1869 y(Xu,)13 b(L.,)f(and)i(Jordan,)f(M.I.)f(\(1993a\),)g (Unsup)q(ervised)j(learning)e(b)o(y)g(EM)g(algorithm)e(based)j(on)f (\014nite)g(mixture)g(of)f(Gaus-)0 1929 y(sians,)h Fa(Pr)n(o)n(c.)19 b(of)c(WCNN'93)p Fs(,)e(P)o(ortland,)g(OR,)g(V)m(ol.)k(I)q(I,)d (431-434.)0 2020 y(Xu,)e(L.,)g(and)g(Jordan,)h(M.I.)e(\(1993b\),)h(EM)h (learning)e(on)i(a)f(generalized)h(\014nite)g(mixture)e(mo)q(del)g(for) h(com)o(bining)e(m)o(ultiple)0 2081 y(classi\014ers,)15 b Fa(Pr)n(o)n(c.)j(of)d(WCNN'93)p Fs(,)e(P)o(ortland,)g(OR,)h(V)m(ol.)i (IV,)e(227-230.)0 2165 y(Xu,)e(L.,)f(and)h(Jordan,)g(M.I.)f(\(1993c\),) g Fa(The)n(or)n(etic)n(al)h(and)i(exp)n(erimental)e(studies)h(of)g(the) g(EM)h(algorithm)e(for)g(unsup)n(ervise)n(d)0 2226 y(le)n(arning)j(b)n (ase)n(d)h(on)g(\014nite)g(Gaussian)g(mixtur)n(es)p Fs(,)e(MIT)g (Computational)e(Cognitiv)o(e)i(Science,)h(T)m(ec)o(hnical)f(Rep)q(ort) h(9302,)0 2286 y(Dept.)j(of)c(Brain)f(and)h(Cognitiv)o(e)f(Science,)i (MIT,)e(Cam)o(bridge,)f(MA.)0 2371 y(Xu,)21 b(L.,)g(Jordan,)g(M.I.)f (and)g(Hin)o(ton,)g(G.E.)f(\(1994\),)i(A)f(Mo)q(di\014ed)g(gating)f (net)o(w)o(ork)i(for)e(the)i(mixtures)e(of)h(exp)q(erts)0 2432 y(arc)o(hitecture,)15 b Fa(Pr)n(o)n(c.)k(of)c(WCNN'94)p Fs(,)e(San)h(Diego,)e(V)m(ol.)17 b(2,)c(405-410.)0 2516 y(Y)m(uille,)19 b(A.)g(L.,)h(Stolorz,)h(P)m(.)e(&)h(Utans,)g(J.)g (\(1994\),)g(Statistical)f(ph)o(ysics,)i(mixtures)e(of)g(distributions) g(and)g(the)i(EM)0 2577 y(algorithm,)11 b Fa(Neur)n(al)j(Computation)h (6)p Fs(,)f(334-340.)952 2817 y Fw(17)p eop %%Page: 18 18 18 17 bop 51 174 a 13985098 13985098 2894397 13024788 36377313 38811238 startTexFig 51 174 a %%BeginDocument: ps/nwcon12b.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 316 260 6111 4708 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 859 4776 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1355 4776 mt (10) s 1971 4615 mt 1971 4561 L 1971 389 mt 1971 443 L 1891 4776 mt (20) s 2507 4615 mt 2507 4561 L 2507 389 mt 2507 443 L 2427 4776 mt (30) s 3043 4615 mt 3043 4561 L 3043 389 mt 3043 443 L 2963 4776 mt (40) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3499 4776 mt (50) s 4114 4615 mt 4114 4561 L 4114 389 mt 4114 443 L 4034 4776 mt (60) s 4650 4615 mt 4650 4561 L 4650 389 mt 4650 443 L 4570 4776 mt (70) s 5186 4615 mt 5186 4561 L 5186 389 mt 5186 443 L 5106 4776 mt (80) s 5722 4615 mt 5722 4561 L 5722 389 mt 5722 443 L 5642 4776 mt (90) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (100) s 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L 504 4668 mt (-5000) s 899 3911 mt 953 3911 L 6258 3911 mt 6204 3911 L 504 3964 mt (-4000) s 899 3206 mt 953 3206 L 6258 3206 mt 6204 3206 L 504 3259 mt (-3000) s 899 2502 mt 953 2502 L 6258 2502 mt 6204 2502 L 504 2555 mt (-2000) s 899 1798 mt 953 1798 L 6258 1798 mt 6204 1798 L 504 1851 mt (-1000) s 899 1093 mt 953 1093 L 6258 1093 mt 6204 1093 L 792 1146 mt (0) s 899 389 mt 953 389 L 6258 389 mt 6204 389 L 552 442 mt (1000) s 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 -1 54 0 54 0 53 -1 54 0 53 -1 54 -1 54 -1 53 -2 54 -1 53 0 54 2 53 11 54 29 54 116 53 -3569 54 3016 53 27 54 -60 54 233 53 37 54 14 53 8 54 4 54 3 53 2 54 1 53 1 54 1 54 1 53 1 54 1 53 1 54 0 53 0 54 -1 54 -1 53 5 54 -376 53 453 953 1108 100 MP stroke DD 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 1 54 1 54 5 53 -132 54 114 53 3 54 0 54 4 53 1 54 0 53 -1 54 0 54 0 53 -1 54 -1 53 -1 54 -1 54 -1 53 -1 54 -2 53 -1 54 -1 53 0 54 0 54 -1 53 -5 54 17 53 0 953 1094 100 MP stroke DA 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 -1 53 0 54 0 53 -1 54 -1 54 -1 53 -1 54 -1 53 0 54 1 53 7 54 18 54 71 53 -2245 54 1917 53 26 54 -21 54 127 53 22 54 10 53 7 54 5 54 4 53 3 54 3 53 2 54 2 54 2 53 2 54 2 53 1 54 1 53 1 54 1 54 1 53 1 54 -89 53 99 953 1095 100 MP stroke gr DA 3009 4936 mt (the learning steps) s 450 3493 mt -90 rotate (the condition number with sign ) s 90 rotate 3043 2344 mt (solid - the original Hessian) s 3043 2555 mt (dash-dot -. the constrained Hessian) s 3043 2766 mt (dashed -- the EM-equivalent Hessian) s SO gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 1012 174 a 13985098 13985098 3749560 11774935 36377313 40521564 startTexFig 1012 174 a %%BeginDocument: ps/nwcon12e.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 471 -48 5956 5235 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 819 4776 mt (20) s 1569 4615 mt 1569 4561 L 1569 389 mt 1569 443 L 1489 4776 mt (30) s 2239 4615 mt 2239 4561 L 2239 389 mt 2239 443 L 2159 4776 mt (40) s 2909 4615 mt 2909 4561 L 2909 389 mt 2909 443 L 2829 4776 mt (50) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3499 4776 mt (60) s 4248 4615 mt 4248 4561 L 4248 389 mt 4248 443 L 4168 4776 mt (70) s 4918 4615 mt 4918 4561 L 4918 389 mt 4918 443 L 4838 4776 mt (80) s 5588 4615 mt 5588 4561 L 5588 389 mt 5588 443 L 5508 4776 mt (90) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (100) s 899 4615 mt 926 4615 L 6258 4615 mt 6231 4615 L 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L /Helvetica 96 FMS /Helvetica 144 FMS 659 4668 mt (10) s /Helvetica 96 FMS 819 4579 mt (0) s /Helvetica 144 FMS 899 4191 mt 926 4191 L 6258 4191 mt 6231 4191 L 899 3943 mt 926 3943 L 6258 3943 mt 6231 3943 L 899 3767 mt 926 3767 L 6258 3767 mt 6231 3767 L 899 3630 mt 926 3630 L 6258 3630 mt 6231 3630 L 899 3519 mt 926 3519 L 6258 3519 mt 6231 3519 L 899 3425 mt 926 3425 L 6258 3425 mt 6231 3425 L 899 3343 mt 926 3343 L 6258 3343 mt 6231 3343 L 899 3271 mt 926 3271 L 6258 3271 mt 6231 3271 L 899 3206 mt 926 3206 L 6258 3206 mt 6231 3206 L 899 3206 mt 953 3206 L 6258 3206 mt 6204 3206 L /Helvetica 96 FMS /Helvetica 144 FMS 659 3259 mt (10) s /Helvetica 96 FMS 819 3170 mt (1) s /Helvetica 144 FMS 899 2782 mt 926 2782 L 6258 2782 mt 6231 2782 L 899 2534 mt 926 2534 L 6258 2534 mt 6231 2534 L 899 2358 mt 926 2358 L 6258 2358 mt 6231 2358 L 899 2222 mt 926 2222 L 6258 2222 mt 6231 2222 L 899 2110 mt 926 2110 L 6258 2110 mt 6231 2110 L 899 2016 mt 926 2016 L 6258 2016 mt 6231 2016 L 899 1934 mt 926 1934 L 6258 1934 mt 6231 1934 L 899 1862 mt 926 1862 L 6258 1862 mt 6231 1862 L 899 1798 mt 926 1798 L 6258 1798 mt 6231 1798 L 899 1798 mt 953 1798 L 6258 1798 mt 6204 1798 L /Helvetica 96 FMS /Helvetica 144 FMS 659 1851 mt (10) s /Helvetica 96 FMS 819 1762 mt (2) s /Helvetica 144 FMS 899 1374 mt 926 1374 L 6258 1374 mt 6231 1374 L 899 1126 mt 926 1126 L 6258 1126 mt 6231 1126 L 899 950 mt 926 950 L 6258 950 mt 6231 950 L 899 813 mt 926 813 L 6258 813 mt 6231 813 L 899 702 mt 926 702 L 6258 702 mt 6231 702 L 899 607 mt 926 607 L 6258 607 mt 6231 607 L 899 526 mt 926 526 L 6258 526 mt 6231 526 L 899 453 mt 926 453 L 6258 453 mt 6231 453 L 899 389 mt 926 389 L 6258 389 mt 6231 389 L 899 389 mt 953 389 L 6258 389 mt 6204 389 L /Helvetica 96 FMS /Helvetica 144 FMS 659 442 mt (10) s /Helvetica 96 FMS 819 353 mt (3) s /Helvetica 144 FMS 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 66 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 -1 67 0 67 0 67 -1 67 0 67 -1 67 -2 67 -1 67 -3 67 -3 67 -5 67 -6 67 -9 67 -11 67 -16 67 -21 67 -27 67 -33 67 -31 67 -9 67 63 67 194 67 355 67 608 1301 1213 75 MP stroke DD 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 66 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 -1 67 0 67 0 67 -1 67 0 67 -1 67 -1 67 -2 67 -3 67 -3 67 -4 67 -6 67 -8 67 -9 67 -12 67 -11 67 -6 67 12 67 54 67 126 67 225 67 421 1301 3061 75 MP stroke DA 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 66 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 0 67 -1 67 0 67 -1 67 0 67 -1 67 -2 67 -1 67 -3 67 -3 67 -5 67 -7 67 -9 67 -12 67 -17 67 -22 67 -29 67 -35 67 -35 67 -15 67 55 67 182 67 339 67 588 1301 1501 75 MP stroke gr DA 3009 4936 mt (the learning steps) s 605 3193 mt -90 rotate (the condition number ) s 90 rotate 2574 2114 mt ( the original Hessian) s 2574 2699 mt ( the constrained Hessian) s 2574 3683 mt ( the EM-equivalent Hessian) s SO gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 435 1128 a Fw(\(a\))959 b(\(b\))0 1239 y(Figure)15 b(1:)20 b(Exp)q(erimen)o(tal)c(results)g(for)e(the)h(estimation)h(of)f(the)g (parameters)f(of)h(a)g(t)o(w)o(o-comp)q(onen)o(t)f(Gaussian)0 1308 y(mixture.)22 b(\(a\))15 b(The)h(condition)h(n)o(um)o(b)q(ers)f (as)g(a)f(function)i(of)e(the)h(iteration)g(n)o(um)o(b)q(er.)22 b(\(b\))16 b(A)g(zo)q(omed)g(v)o(ersion)0 1376 y(of)k(\(a\))f(after)h (discarding)i(the)e(\014rst)g(25)g(iterations.)35 b(The)21 b(terminology)g(`original,)h(constrained,)g(and)e(EM-)0 1445 y(equiv)m(alen)o(t)d(Hessians')e(refers)g(to)g(the)g(matrices)g Fo(H)q(;)8 b(E)948 1429 y Fn(T)974 1445 y Fo(H)t(E)s(;)g(E)1111 1429 y Fn(T)1136 1445 y Fo(P)e(H)t(E)17 b Fw(resp)q(ectiv)o(ely)l(.)51 1528 y 13985098 13985098 2894397 13024788 36377313 38811238 startTexFig 51 1528 a %%BeginDocument: ps/nwcon1b.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 316 260 6111 4708 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 859 4776 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1355 4776 mt (50) s 1971 4615 mt 1971 4561 L 1971 389 mt 1971 443 L 1851 4776 mt (100) s 2507 4615 mt 2507 4561 L 2507 389 mt 2507 443 L 2387 4776 mt (150) s 3043 4615 mt 3043 4561 L 3043 389 mt 3043 443 L 2923 4776 mt (200) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3459 4776 mt (250) s 4114 4615 mt 4114 4561 L 4114 389 mt 4114 443 L 3994 4776 mt (300) s 4650 4615 mt 4650 4561 L 4650 389 mt 4650 443 L 4530 4776 mt (350) s 5186 4615 mt 5186 4561 L 5186 389 mt 5186 443 L 5066 4776 mt (400) s 5722 4615 mt 5722 4561 L 5722 389 mt 5722 443 L 5602 4776 mt (450) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (500) s 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L 504 4668 mt (-1000) s 899 4192 mt 953 4192 L 6258 4192 mt 6204 4192 L 584 4245 mt (-800) s 899 3770 mt 953 3770 L 6258 3770 mt 6204 3770 L 584 3823 mt (-600) s 899 3347 mt 953 3347 L 6258 3347 mt 6204 3347 L 584 3400 mt (-400) s 899 2925 mt 953 2925 L 6258 2925 mt 6204 2925 L 584 2978 mt (-200) s 899 2502 mt 953 2502 L 6258 2502 mt 6204 2502 L 792 2555 mt (0) s 899 2079 mt 953 2079 L 6258 2079 mt 6204 2079 L 632 2132 mt (200) s 899 1657 mt 953 1657 L 6258 1657 mt 6204 1657 L 632 1710 mt (400) s 899 1234 mt 953 1234 L 6258 1234 mt 6204 1234 L 632 1287 mt (600) s 899 812 mt 953 812 L 6258 812 mt 6204 812 L 632 865 mt (800) s 899 389 mt 953 389 L 6258 389 mt 6204 389 L 552 442 mt (1000) s 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 5197 1760 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 4136 1767 100 MP stroke 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 -1 11 0 11 0 11 -1 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 -1 10 0 11 -1 11 0 11 -1 10 -1 11 0 11 -1 10 -1 11 0 11 -1 11 -1 10 0 3075 1803 100 MP stroke 11 -1 11 -1 10 -1 11 0 11 -1 11 -1 10 -1 11 0 11 -1 10 -1 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -2 11 -1 11 -1 11 -1 10 -1 11 -1 11 -2 10 -1 11 -1 11 -1 11 -2 10 -1 11 -1 11 -2 10 -1 11 -1 11 -2 11 -1 10 -1 11 -2 11 -1 10 -2 11 -1 11 -2 11 -1 10 -1 11 -2 11 -1 10 -2 11 -2 11 -1 11 -2 10 -1 11 -2 11 -1 11 -2 10 -2 11 -1 11 -2 10 -1 11 -2 11 -2 11 -1 10 -2 11 -1 11 -2 10 -2 11 -1 11 -2 11 -2 10 -1 11 -2 11 -1 10 -2 11 -2 11 -1 11 -2 10 -1 11 -2 11 -1 10 -2 11 -1 11 -2 11 -1 10 -1 2014 1929 100 MP stroke 11 -2 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 1 11 1 11 1 10 1 11 2 11 2 10 2 11 2 11 3 11 3 10 4 11 3 11 4 10 5 11 5 11 5 11 6 10 6 11 7 11 8 10 8 11 8 11 10 11 10 10 11 11 12 11 13 11 13 10 15 11 16 11 16 10 18 11 19 11 19 11 20 10 20 11 19 11 16 10 11 11 3 11 -8 11 -19 10 -28 11 -16 11 855 10 -3613 11 1166 11 209 11 74 10 30 11 9 11 -1 10 -8 11 -13 11 -14 11 -14 10 -15 11 -12 11 -11 10 -7 11 -4 11 1 11 5 10 10 11 13 11 17 11 19 10 21 11 22 11 22 10 20 11 17 11 15 11 12 10 8 11 6 11 5 10 9 953 2811 100 MP stroke 11 21 11 33 11 -1346 10 1558 910 2545 5 MP stroke DD 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 2373 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 4136 2374 100 MP stroke 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 3075 2382 100 MP stroke 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 -1 11 0 11 0 11 -1 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 2014 2412 100 MP stroke 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 11 0 10 -1 11 0 11 -1 10 0 11 0 11 -1 11 0 10 -1 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 -1 11 -3 11 -3 11 -4 10 -4 11 -1 11 27 10 -151 11 42 11 7 11 2 10 1 11 0 11 0 10 -1 11 -1 11 -1 11 -1 10 -2 11 -1 11 -3 10 -2 11 -3 11 -4 11 -4 10 -5 11 -6 11 -6 11 -6 10 -6 11 -5 11 -5 10 -3 11 -1 11 -1 11 0 10 0 11 -2 11 -4 10 -9 953 2594 100 MP stroke 11 -9 11 25 11 74 10 -1 910 2505 5 MP stroke DA 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 2046 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 4136 2051 100 MP stroke 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 -1 11 0 10 0 11 -1 11 0 11 0 10 0 11 -1 11 0 10 0 11 -1 11 0 11 0 10 -1 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 -1 11 0 11 0 10 -1 11 0 11 -1 11 0 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 3075 2076 100 MP stroke 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 -1 11 0 10 -1 11 0 11 -1 11 -1 10 0 11 -1 11 -1 10 0 11 -1 11 -1 11 0 10 -1 11 -1 11 0 11 -1 10 -1 11 -1 11 0 10 -1 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 10 -1 11 -1 11 -1 11 0 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -2 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -2 11 -1 11 -1 10 -1 11 -1 11 -1 11 -2 10 -1 11 -1 11 -1 10 -1 11 -1 11 -2 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -2 10 -1 2014 2164 100 MP stroke 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 11 -1 10 -1 11 0 11 -1 10 -1 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 0 10 1 11 0 11 1 10 0 11 1 11 1 11 1 10 1 11 2 11 2 10 1 11 2 11 3 11 2 10 3 11 3 11 4 10 3 11 4 11 5 11 4 10 5 11 6 11 6 11 6 10 7 11 7 11 8 10 9 11 8 11 9 11 10 10 9 11 8 11 6 10 3 11 -2 11 -7 11 -15 10 -20 11 -14 11 414 10 -1764 11 575 11 105 11 39 10 18 11 8 11 3 10 1 11 -1 11 -2 11 -2 10 -1 11 -1 11 -1 10 2 11 2 11 4 11 5 10 6 11 8 11 9 11 9 10 9 11 10 11 9 10 8 11 7 11 6 11 5 10 4 11 3 11 3 10 3 953 2582 100 MP stroke 11 7 11 9 11 -894 10 952 910 2508 5 MP stroke gr DA 3009 4936 mt (the learning steps) s 450 3493 mt -90 rotate (the condition number with sign ) s 90 rotate 3043 970 mt (solid - the original Hessian) s 3043 1287 mt (dash-dot -. the constrained Hessian) s 3043 1604 mt (dashed -- the EM-equivalent Hessian) s SO gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 1012 1528 a 13985098 13985098 3749560 11774935 36377313 40521564 startTexFig 1012 1528 a %%BeginDocument: ps/nwcon1e.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 471 -48 5956 5235 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 859 4776 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1355 4776 mt (50) s 1971 4615 mt 1971 4561 L 1971 389 mt 1971 443 L 1851 4776 mt (100) s 2507 4615 mt 2507 4561 L 2507 389 mt 2507 443 L 2387 4776 mt (150) s 3043 4615 mt 3043 4561 L 3043 389 mt 3043 443 L 2923 4776 mt (200) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3459 4776 mt (250) s 4114 4615 mt 4114 4561 L 4114 389 mt 4114 443 L 3994 4776 mt (300) s 4650 4615 mt 4650 4561 L 4650 389 mt 4650 443 L 4530 4776 mt (350) s 5186 4615 mt 5186 4561 L 5186 389 mt 5186 443 L 5066 4776 mt (400) s 5722 4615 mt 5722 4561 L 5722 389 mt 5722 443 L 5602 4776 mt (450) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (500) s 899 4615 mt 926 4615 L 6258 4615 mt 6231 4615 L 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L /Helvetica 96 FMS /Helvetica 144 FMS 659 4668 mt (10) s /Helvetica 96 FMS 819 4579 mt (1) s /Helvetica 144 FMS 899 3979 mt 926 3979 L 6258 3979 mt 6231 3979 L 899 3607 mt 926 3607 L 6258 3607 mt 6231 3607 L 899 3343 mt 926 3343 L 6258 3343 mt 6231 3343 L 899 3138 mt 926 3138 L 6258 3138 mt 6231 3138 L 899 2971 mt 926 2971 L 6258 2971 mt 6231 2971 L 899 2829 mt 926 2829 L 6258 2829 mt 6231 2829 L 899 2707 mt 926 2707 L 6258 2707 mt 6231 2707 L 899 2599 mt 926 2599 L 6258 2599 mt 6231 2599 L 899 2502 mt 926 2502 L 6258 2502 mt 6231 2502 L 899 2502 mt 953 2502 L 6258 2502 mt 6204 2502 L /Helvetica 96 FMS /Helvetica 144 FMS 659 2555 mt (10) s /Helvetica 96 FMS 819 2466 mt (2) s /Helvetica 144 FMS 899 1866 mt 926 1866 L 6258 1866 mt 6231 1866 L 899 1494 mt 926 1494 L 6258 1494 mt 6231 1494 L 899 1230 mt 926 1230 L 6258 1230 mt 6231 1230 L 899 1025 mt 926 1025 L 6258 1025 mt 6231 1025 L 899 858 mt 926 858 L 6258 858 mt 6231 858 L 899 716 mt 926 716 L 6258 716 mt 6231 716 L 899 594 mt 926 594 L 6258 594 mt 6231 594 L 899 486 mt 926 486 L 6258 486 mt 6231 486 L 899 389 mt 926 389 L 6258 389 mt 6231 389 L 899 389 mt 953 389 L 6258 389 mt 6204 389 L /Helvetica 96 FMS /Helvetica 144 FMS 659 442 mt (10) s /Helvetica 96 FMS 819 353 mt (3) s /Helvetica 144 FMS 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 1349 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 4136 1358 100 MP stroke 11 0 11 -1 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 -1 11 0 11 0 10 -1 11 0 11 0 10 -1 11 0 11 -1 11 0 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 10 -1 11 -1 11 0 11 -1 10 0 11 -1 11 -1 10 0 11 -1 11 -1 11 0 10 -1 11 -1 11 -1 10 0 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 11 -1 10 -1 11 0 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 3075 1405 100 MP stroke 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -2 10 -1 11 -1 11 -1 11 -2 10 -1 11 -1 11 -1 10 -2 11 -1 11 -2 11 -1 10 -1 11 -2 11 -1 11 -2 10 -1 11 -2 11 -1 10 -2 11 -1 11 -2 11 -2 10 -1 11 -2 11 -2 10 -1 11 -2 11 -2 11 -2 10 -2 11 -1 11 -2 10 -2 11 -2 11 -2 11 -2 10 -2 11 -2 11 -2 10 -2 11 -2 11 -2 11 -2 10 -2 11 -3 11 -2 10 -2 11 -2 11 -2 11 -3 10 -2 11 -2 11 -3 11 -2 10 -2 11 -3 11 -2 10 -3 11 -2 11 -3 11 -2 10 -2 11 -3 11 -2 10 -3 11 -2 11 -3 11 -2 10 -3 11 -2 11 -3 10 -2 11 -3 11 -2 11 -3 10 -2 11 -2 11 -3 10 -2 11 -2 11 -3 11 -2 10 -2 2014 1587 100 MP stroke 11 -2 11 -2 10 -2 11 -2 11 -2 11 -1 10 -2 11 -1 11 -2 11 -1 10 -1 11 -1 11 -1 10 0 11 -1 11 0 11 0 10 0 11 1 11 0 10 1 11 2 11 1 11 2 10 2 11 3 11 3 10 3 11 4 11 5 11 4 10 6 11 5 11 7 10 7 11 7 11 8 11 9 10 10 11 10 11 11 10 11 11 13 11 13 11 14 10 15 11 16 11 17 11 18 10 18 11 20 11 20 10 21 11 22 11 22 11 22 10 21 11 20 11 16 10 11 11 4 11 -8 11 -19 10 -30 11 -18 11 646 1306 592 67 MP stroke DD 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 5197 2954 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 4136 2965 100 MP stroke 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 -1 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 -1 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 11 -1 10 -1 11 0 11 -1 10 -1 11 0 11 -1 11 -1 10 0 11 -1 11 -1 10 0 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 10 -1 11 -1 11 0 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 3075 3022 100 MP stroke 11 -2 11 -1 10 -1 11 -1 11 -1 11 -2 10 -1 11 -1 11 -2 10 -1 11 -1 11 -2 11 -1 10 -2 11 -1 11 -2 10 -1 11 -2 11 -1 11 -2 10 -2 11 -1 11 -2 10 -2 11 -1 11 -2 11 -2 10 -2 11 -2 11 -1 11 -2 10 -2 11 -2 11 -2 10 -2 11 -2 11 -2 11 -2 10 -3 11 -2 11 -2 10 -2 11 -3 11 -2 11 -2 10 -3 11 -2 11 -2 10 -3 11 -3 11 -2 11 -3 10 -2 11 -3 11 -3 10 -3 11 -2 11 -3 11 -3 10 -3 11 -3 11 -3 10 -3 11 -3 11 -3 11 -4 10 -3 11 -3 11 -3 11 -4 10 -3 11 -4 11 -3 10 -4 11 -3 11 -4 11 -3 10 -4 11 -4 11 -4 10 -4 11 -3 11 -4 11 -4 10 -4 11 -4 11 -4 10 -5 11 -4 11 -4 11 -4 10 -4 11 -5 11 -4 10 -4 11 -5 11 -4 11 -5 10 -4 2014 3286 100 MP stroke 11 -4 11 -5 10 -4 11 -5 11 -4 11 -5 10 -4 11 -5 11 -4 11 -5 10 -4 11 -5 11 -4 10 -4 11 -5 11 -4 11 -4 10 -4 11 -4 11 -4 10 -4 11 -4 11 -4 11 -3 10 -4 11 -3 11 -3 10 -4 11 -2 11 -3 11 -3 10 -2 11 -2 11 -2 10 -2 11 -2 11 -1 11 -1 10 -1 11 0 11 -1 10 0 11 1 11 0 11 1 10 1 11 1 11 2 11 2 10 2 11 3 11 2 10 2 11 3 11 1 11 1 10 -1 11 -5 11 -9 10 -17 11 -27 11 -40 11 -52 10 -55 11 -21 11 329 1306 3300 67 MP stroke DA 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 1796 100 MP stroke 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 11 0 10 0 11 -1 11 0 10 0 11 0 11 0 11 0 10 -1 11 0 11 0 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 0 11 -1 4136 1807 100 MP stroke 11 0 11 0 10 0 11 0 11 -1 10 0 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 0 10 -1 11 0 11 0 11 -1 10 0 11 0 11 -1 10 0 11 0 11 -1 11 0 10 0 11 -1 11 0 10 -1 11 0 11 0 11 -1 10 0 11 -1 11 0 10 0 11 -1 11 0 11 -1 10 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 0 11 -1 11 0 10 -1 11 0 11 -1 10 0 11 -1 11 -1 11 0 10 -1 11 0 11 -1 10 -1 11 0 11 -1 11 -1 10 0 11 -1 11 -1 10 0 11 -1 11 -1 11 -1 10 0 11 -1 11 -1 10 -1 11 -1 11 -1 11 0 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 3075 1858 100 MP stroke 11 -1 11 -1 10 -1 11 -1 11 -1 11 -1 10 -1 11 -1 11 -2 10 -1 11 -1 11 -1 11 -1 10 -2 11 -1 11 -1 10 -2 11 -1 11 -1 11 -2 10 -1 11 -2 11 -1 10 -2 11 -1 11 -2 11 -1 10 -2 11 -1 11 -2 11 -2 10 -1 11 -2 11 -2 10 -1 11 -2 11 -2 11 -2 10 -2 11 -2 11 -1 10 -2 11 -2 11 -2 11 -2 10 -2 11 -2 11 -3 10 -2 11 -2 11 -2 11 -2 10 -3 11 -2 11 -2 10 -2 11 -3 11 -2 11 -3 10 -2 11 -3 11 -2 10 -3 11 -2 11 -3 11 -2 10 -3 11 -3 11 -2 11 -3 10 -3 11 -3 11 -2 10 -3 11 -3 11 -3 11 -3 10 -3 11 -3 11 -3 10 -3 11 -3 11 -3 11 -3 10 -3 11 -3 11 -3 10 -3 11 -3 11 -3 11 -3 10 -3 11 -3 11 -3 10 -3 11 -3 11 -3 11 -3 10 -3 2014 2070 100 MP stroke 11 -3 11 -3 10 -2 11 -3 11 -3 11 -2 10 -3 11 -2 11 -2 11 -3 10 -2 11 -2 11 -1 10 -2 11 -2 11 -1 11 -1 10 -1 11 -1 11 0 10 -1 11 0 11 1 11 0 10 1 11 1 11 2 10 2 11 2 11 3 11 3 10 3 11 4 11 5 10 5 11 6 11 6 11 7 10 7 11 8 11 9 10 10 11 10 11 11 11 12 10 12 11 14 11 14 11 15 10 16 11 17 11 18 10 18 11 19 11 19 11 19 10 18 11 16 11 12 10 6 11 -3 11 -15 11 -29 10 -41 11 -30 11 632 1306 1245 67 MP stroke gr DA 3009 4936 mt (the learning steps) s 605 3193 mt -90 rotate (the condition number ) s 90 rotate 3579 1547 mt (solid - the original Hessian) s 3579 1966 mt (the constrained Hessian) s 3579 2882 mt (the EM-equivalent Hessian) s SO gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 435 2482 a Fw(\(a\))959 b(\(b\))0 2593 y(Figure)15 b(2:)20 b(Exp)q(erimen)o(tal)c(results)g(for)e(the)h(estimation)h(of)f(the)g (parameters)f(of)h(a)g(t)o(w)o(o-comp)q(onen)o(t)f(Gaussian)0 2662 y(mixture)i(\(cf.)j(Fig.)h(1\).)f(The)d(separation)f(of)g(the)g (Gaussians)g(is)h(half)f(the)h(separation)f(in)h(Fig.)k(1.)952 2817 y(18)p eop %%Page: 19 19 19 18 bop 51 817 a 13985098 13985098 3552215 11774935 36377313 40521564 startTexFig 51 817 a %%BeginDocument: ps/nwevl2.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 439 -48 5988 5235 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 859 4776 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1355 4776 mt (10) s 1971 4615 mt 1971 4561 L 1971 389 mt 1971 443 L 1891 4776 mt (20) s 2507 4615 mt 2507 4561 L 2507 389 mt 2507 443 L 2427 4776 mt (30) s 3043 4615 mt 3043 4561 L 3043 389 mt 3043 443 L 2963 4776 mt (40) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3499 4776 mt (50) s 4114 4615 mt 4114 4561 L 4114 389 mt 4114 443 L 4034 4776 mt (60) s 4650 4615 mt 4650 4561 L 4650 389 mt 4650 443 L 4570 4776 mt (70) s 5186 4615 mt 5186 4561 L 5186 389 mt 5186 443 L 5106 4776 mt (80) s 5722 4615 mt 5722 4561 L 5722 389 mt 5722 443 L 5642 4776 mt (90) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (100) s 899 4615 mt 926 4615 L 6258 4615 mt 6231 4615 L 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L /Helvetica 96 FMS /Helvetica 144 FMS 627 4668 mt (10) s /Helvetica 96 FMS 787 4579 mt (-1) s /Helvetica 144 FMS 899 4361 mt 926 4361 L 6258 4361 mt 6231 4361 L 899 4212 mt 926 4212 L 6258 4212 mt 6231 4212 L 899 4106 mt 926 4106 L 6258 4106 mt 6231 4106 L 899 4024 mt 926 4024 L 6258 4024 mt 6231 4024 L 899 3957 mt 926 3957 L 6258 3957 mt 6231 3957 L 899 3901 mt 926 3901 L 6258 3901 mt 6231 3901 L 899 3852 mt 926 3852 L 6258 3852 mt 6231 3852 L 899 3808 mt 926 3808 L 6258 3808 mt 6231 3808 L 899 3770 mt 926 3770 L 6258 3770 mt 6231 3770 L 899 3770 mt 953 3770 L 6258 3770 mt 6204 3770 L /Helvetica 96 FMS /Helvetica 144 FMS 627 3823 mt (10) s /Helvetica 96 FMS 787 3734 mt (0) s /Helvetica 144 FMS 899 3515 mt 926 3515 L 6258 3515 mt 6231 3515 L 899 3367 mt 926 3367 L 6258 3367 mt 6231 3367 L 899 3261 mt 926 3261 L 6258 3261 mt 6231 3261 L 899 3179 mt 926 3179 L 6258 3179 mt 6231 3179 L 899 3112 mt 926 3112 L 6258 3112 mt 6231 3112 L 899 3056 mt 926 3056 L 6258 3056 mt 6231 3056 L 899 3007 mt 926 3007 L 6258 3007 mt 6231 3007 L 899 2963 mt 926 2963 L 6258 2963 mt 6231 2963 L 899 2925 mt 926 2925 L 6258 2925 mt 6231 2925 L 899 2925 mt 953 2925 L 6258 2925 mt 6204 2925 L /Helvetica 96 FMS /Helvetica 144 FMS 627 2978 mt (10) s /Helvetica 96 FMS 787 2889 mt (1) s /Helvetica 144 FMS 899 2670 mt 926 2670 L 6258 2670 mt 6231 2670 L 899 2521 mt 926 2521 L 6258 2521 mt 6231 2521 L 899 2416 mt 926 2416 L 6258 2416 mt 6231 2416 L 899 2334 mt 926 2334 L 6258 2334 mt 6231 2334 L 899 2267 mt 926 2267 L 6258 2267 mt 6231 2267 L 899 2210 mt 926 2210 L 6258 2210 mt 6231 2210 L 899 2161 mt 926 2161 L 6258 2161 mt 6231 2161 L 899 2118 mt 926 2118 L 6258 2118 mt 6231 2118 L 899 2079 mt 926 2079 L 6258 2079 mt 6231 2079 L 899 2079 mt 953 2079 L 6258 2079 mt 6204 2079 L /Helvetica 96 FMS /Helvetica 144 FMS 627 2132 mt (10) s /Helvetica 96 FMS 787 2043 mt (2) s /Helvetica 144 FMS 899 1825 mt 926 1825 L 6258 1825 mt 6231 1825 L 899 1676 mt 926 1676 L 6258 1676 mt 6231 1676 L 899 1571 mt 926 1571 L 6258 1571 mt 6231 1571 L 899 1489 mt 926 1489 L 6258 1489 mt 6231 1489 L 899 1422 mt 926 1422 L 6258 1422 mt 6231 1422 L 899 1365 mt 926 1365 L 6258 1365 mt 6231 1365 L 899 1316 mt 926 1316 L 6258 1316 mt 6231 1316 L 899 1273 mt 926 1273 L 6258 1273 mt 6231 1273 L 899 1234 mt 926 1234 L 6258 1234 mt 6231 1234 L 899 1234 mt 953 1234 L 6258 1234 mt 6204 1234 L /Helvetica 96 FMS /Helvetica 144 FMS 627 1287 mt (10) s /Helvetica 96 FMS 787 1198 mt (3) s /Helvetica 144 FMS 899 980 mt 926 980 L 6258 980 mt 6231 980 L 899 831 mt 926 831 L 6258 831 mt 6231 831 L 899 725 mt 926 725 L 6258 725 mt 6231 725 L 899 643 mt 926 643 L 6258 643 mt 6231 643 L 899 577 mt 926 577 L 6258 577 mt 6231 577 L 899 520 mt 926 520 L 6258 520 mt 6231 520 L 899 471 mt 926 471 L 6258 471 mt 6231 471 L 899 428 mt 926 428 L 6258 428 mt 6231 428 L 899 389 mt 926 389 L 6258 389 mt 6231 389 L 899 389 mt 953 389 L 6258 389 mt 6204 389 L /Helvetica 96 FMS /Helvetica 144 FMS 627 442 mt (10) s /Helvetica 96 FMS 787 353 mt (4) s /Helvetica 144 FMS 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 1 54 0 53 0 54 0 53 0 54 1 54 0 53 1 54 1 53 1 54 2 54 3 53 4 54 5 53 8 54 12 53 17 54 22 54 30 53 37 54 40 53 44 54 40 54 -10 53 -69 54 -74 53 -58 54 -41 54 -31 53 -24 54 -19 53 -16 54 -13 54 -12 53 -10 54 -9 53 -9 54 -7 53 -7 54 -6 54 -4 53 -5 54 -94 53 -18 953 1079 100 MP stroke DD 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 1 53 0 54 0 54 0 53 1 54 1 53 1 54 4 53 5 54 5 54 4 53 2 54 2 53 3 54 3 54 -7 53 -11 54 -6 53 -3 54 -2 54 -1 53 0 54 -1 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 -1 53 133 953 3637 100 MP stroke DA 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 0 54 0 54 0 53 0 54 0 53 1 54 0 54 0 53 1 54 0 53 1 54 1 54 2 53 2 54 4 53 4 54 7 53 10 54 13 54 18 53 21 54 24 53 23 54 16 54 -35 53 -94 54 -102 53 -87 54 -73 54 -61 53 -53 54 -47 53 -43 54 -41 54 -39 53 -38 54 -37 53 -36 54 -36 53 -34 54 -32 54 -29 53 -27 54 -168 53 -96 953 2000 100 MP stroke SO 53 0 953 4615 2 MP stroke gr 3009 4936 mt (the learning steps) s 573 3298 mt -90 rotate (the maximum eigenvalue) s 90 rotate 3311 696 mt ( the original Hessian) s 3311 1138 mt ( the constrained Hessian) s 3311 3568 mt ( the EM-equivalent Hessian) s gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 1012 817 a 13985098 13985098 3552215 11774935 36377313 40521564 startTexFig 1012 817 a %%BeginDocument: ps/nwevl1.ps /MathWorks 120 dict begin /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef /portraitMode 0 def /landscapeMode 1 def /dpi2point 0 def /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 1 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 439 -48 5988 5235 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS /Helvetica 96 FMS /Helvetica 144 FMS c1 0 0 6917 5187 PR 6 w DO 4 w SO 6 w c0 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 4615 mt 899 4615 L 6258 4615 mt 6258 4615 L 899 4615 mt 6258 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4615 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L 859 4776 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1355 4776 mt (50) s 1971 4615 mt 1971 4561 L 1971 389 mt 1971 443 L 1851 4776 mt (100) s 2507 4615 mt 2507 4561 L 2507 389 mt 2507 443 L 2387 4776 mt (150) s 3043 4615 mt 3043 4561 L 3043 389 mt 3043 443 L 2923 4776 mt (200) s 3579 4615 mt 3579 4561 L 3579 389 mt 3579 443 L 3459 4776 mt (250) s 4114 4615 mt 4114 4561 L 4114 389 mt 4114 443 L 3994 4776 mt (300) s 4650 4615 mt 4650 4561 L 4650 389 mt 4650 443 L 4530 4776 mt (350) s 5186 4615 mt 5186 4561 L 5186 389 mt 5186 443 L 5066 4776 mt (400) s 5722 4615 mt 5722 4561 L 5722 389 mt 5722 443 L 5602 4776 mt (450) s 6258 4615 mt 6258 4561 L 6258 389 mt 6258 443 L 6138 4776 mt (500) s 899 4615 mt 926 4615 L 6258 4615 mt 6231 4615 L 899 4615 mt 953 4615 L 6258 4615 mt 6204 4615 L /Helvetica 96 FMS /Helvetica 144 FMS 627 4668 mt (10) s /Helvetica 96 FMS 787 4579 mt (-1) s /Helvetica 144 FMS 899 4361 mt 926 4361 L 6258 4361 mt 6231 4361 L 899 4212 mt 926 4212 L 6258 4212 mt 6231 4212 L 899 4106 mt 926 4106 L 6258 4106 mt 6231 4106 L 899 4024 mt 926 4024 L 6258 4024 mt 6231 4024 L 899 3957 mt 926 3957 L 6258 3957 mt 6231 3957 L 899 3901 mt 926 3901 L 6258 3901 mt 6231 3901 L 899 3852 mt 926 3852 L 6258 3852 mt 6231 3852 L 899 3808 mt 926 3808 L 6258 3808 mt 6231 3808 L 899 3770 mt 926 3770 L 6258 3770 mt 6231 3770 L 899 3770 mt 953 3770 L 6258 3770 mt 6204 3770 L /Helvetica 96 FMS /Helvetica 144 FMS 627 3823 mt (10) s /Helvetica 96 FMS 787 3734 mt (0) s /Helvetica 144 FMS 899 3515 mt 926 3515 L 6258 3515 mt 6231 3515 L 899 3367 mt 926 3367 L 6258 3367 mt 6231 3367 L 899 3261 mt 926 3261 L 6258 3261 mt 6231 3261 L 899 3179 mt 926 3179 L 6258 3179 mt 6231 3179 L 899 3112 mt 926 3112 L 6258 3112 mt 6231 3112 L 899 3056 mt 926 3056 L 6258 3056 mt 6231 3056 L 899 3007 mt 926 3007 L 6258 3007 mt 6231 3007 L 899 2963 mt 926 2963 L 6258 2963 mt 6231 2963 L 899 2925 mt 926 2925 L 6258 2925 mt 6231 2925 L 899 2925 mt 953 2925 L 6258 2925 mt 6204 2925 L /Helvetica 96 FMS /Helvetica 144 FMS 627 2978 mt (10) s /Helvetica 96 FMS 787 2889 mt (1) s /Helvetica 144 FMS 899 2670 mt 926 2670 L 6258 2670 mt 6231 2670 L 899 2521 mt 926 2521 L 6258 2521 mt 6231 2521 L 899 2416 mt 926 2416 L 6258 2416 mt 6231 2416 L 899 2334 mt 926 2334 L 6258 2334 mt 6231 2334 L 899 2267 mt 926 2267 L 6258 2267 mt 6231 2267 L 899 2210 mt 926 2210 L 6258 2210 mt 6231 2210 L 899 2161 mt 926 2161 L 6258 2161 mt 6231 2161 L 899 2118 mt 926 2118 L 6258 2118 mt 6231 2118 L 899 2079 mt 926 2079 L 6258 2079 mt 6231 2079 L 899 2079 mt 953 2079 L 6258 2079 mt 6204 2079 L /Helvetica 96 FMS /Helvetica 144 FMS 627 2132 mt (10) s /Helvetica 96 FMS 787 2043 mt (2) s /Helvetica 144 FMS 899 1825 mt 926 1825 L 6258 1825 mt 6231 1825 L 899 1676 mt 926 1676 L 6258 1676 mt 6231 1676 L 899 1571 mt 926 1571 L 6258 1571 mt 6231 1571 L 899 1489 mt 926 1489 L 6258 1489 mt 6231 1489 L 899 1422 mt 926 1422 L 6258 1422 mt 6231 1422 L 899 1365 mt 926 1365 L 6258 1365 mt 6231 1365 L 899 1316 mt 926 1316 L 6258 1316 mt 6231 1316 L 899 1273 mt 926 1273 L 6258 1273 mt 6231 1273 L 899 1234 mt 926 1234 L 6258 1234 mt 6231 1234 L 899 1234 mt 953 1234 L 6258 1234 mt 6204 1234 L /Helvetica 96 FMS /Helvetica 144 FMS 627 1287 mt (10) s /Helvetica 96 FMS 787 1198 mt (3) s /Helvetica 144 FMS 899 980 mt 926 980 L 6258 980 mt 6231 980 L 899 831 mt 926 831 L 6258 831 mt 6231 831 L 899 725 mt 926 725 L 6258 725 mt 6231 725 L 899 643 mt 926 643 L 6258 643 mt 6231 643 L 899 577 mt 926 577 L 6258 577 mt 6231 577 L 899 520 mt 926 520 L 6258 520 mt 6231 520 L 899 471 mt 926 471 L 6258 471 mt 6231 471 L 899 428 mt 926 428 L 6258 428 mt 6231 428 L 899 389 mt 926 389 L 6258 389 mt 6231 389 L 899 389 mt 953 389 L 6258 389 mt 6204 389 L /Helvetica 96 FMS /Helvetica 144 FMS 627 442 mt (10) s /Helvetica 96 FMS 787 353 mt (4) s /Helvetica 144 FMS 899 4615 mt 6258 4615 L 899 389 mt 6258 389 L 899 4615 mt 899 389 L 6258 4615 mt 6258 389 L 899 389 mt 899 389 L 6258 389 mt 6258 389 L gs 899 389 5360 4227 MR c np 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 846 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 1 4136 844 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 3075 838 100 MP stroke 11 0 11 0 10 1 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 10 0 11 1 11 0 11 0 10 0 11 1 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 1 11 0 10 0 11 1 11 0 10 0 11 1 11 0 11 0 10 1 11 0 11 0 10 1 11 0 11 0 11 1 10 0 11 0 11 1 10 0 11 1 11 0 11 1 10 0 11 0 11 1 10 0 11 1 11 0 11 1 10 0 11 1 11 0 11 1 10 1 11 0 11 1 10 0 11 1 11 1 11 0 10 1 11 1 11 0 10 1 11 1 11 0 11 1 10 1 11 1 11 0 10 1 11 1 11 1 11 1 10 1 11 1 11 1 10 1 11 1 11 1 11 1 10 1 2014 794 100 MP stroke 11 1 11 1 10 1 11 2 11 1 11 1 10 1 11 2 11 1 11 2 10 1 11 1 11 2 10 2 11 1 11 2 11 2 10 1 11 2 11 2 10 2 11 2 11 2 11 2 10 2 11 3 11 2 10 2 11 3 11 2 11 3 10 3 11 3 11 3 10 3 11 3 11 3 11 3 10 4 11 3 11 4 10 4 11 4 11 4 11 4 10 5 11 4 11 5 11 5 10 5 11 5 11 6 10 5 11 6 11 7 11 6 10 6 11 7 11 7 10 6 11 7 11 6 11 5 10 3 11 0 11 -6 10 -11 11 -17 11 -22 11 -23 10 -24 11 -23 11 -22 10 -19 11 -17 11 -16 11 -14 10 -12 11 -12 11 -10 10 -10 11 -9 11 -9 11 -8 10 -8 11 -8 11 -7 11 -7 10 -7 11 -7 11 -6 10 -6 11 -6 11 -6 11 -5 10 -5 11 -4 11 -4 10 -3 953 959 100 MP stroke 11 -2 11 -2 11 -141 10 24 910 1080 5 MP stroke DD 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 3770 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 4136 3770 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 3075 3770 100 MP stroke 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 2014 3769 100 MP stroke 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 1 11 0 11 0 11 1 10 0 11 1 11 0 10 -1 11 0 11 -1 11 -1 10 0 11 -1 11 0 10 0 11 -1 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 -1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 953 3770 100 MP stroke 11 0 11 1 11 -1 10 147 910 3623 5 MP stroke DA 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 5197 1025 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 4136 1024 100 MP stroke 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 1 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 1 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 0 11 0 10 0 11 0 11 1 10 0 11 0 11 0 11 0 10 0 3075 1019 100 MP stroke 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 10 0 11 0 11 0 11 1 10 0 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 11 0 10 1 11 0 11 0 10 0 11 0 11 1 11 0 10 0 11 0 11 0 10 1 11 0 11 0 11 0 10 1 11 0 11 0 10 0 11 1 11 0 11 0 10 0 11 1 11 0 10 0 11 1 11 0 11 0 10 1 11 0 11 0 10 1 11 0 11 0 11 1 10 0 11 0 11 1 11 0 10 1 11 0 11 0 10 1 11 0 11 1 11 0 10 1 11 0 11 1 10 0 11 1 11 0 11 1 10 0 11 1 11 1 10 0 11 1 11 0 11 1 10 1 11 0 11 1 10 1 11 1 11 0 11 1 10 1 2014 987 100 MP stroke 11 1 11 1 10 0 11 1 11 1 11 1 10 1 11 1 11 1 11 1 10 1 11 1 11 1 10 2 11 1 11 1 11 1 10 2 11 1 11 1 10 2 11 1 11 2 11 1 10 2 11 2 11 1 10 2 11 2 11 2 11 2 10 2 11 2 11 2 10 3 11 2 11 2 11 3 10 3 11 2 11 3 10 3 11 3 11 4 11 3 10 4 11 3 11 4 11 4 10 4 11 4 11 5 10 4 11 5 11 5 11 5 10 5 11 6 11 5 10 5 11 4 11 3 11 1 10 -1 11 -6 11 -11 10 -18 11 -24 11 -29 11 -32 10 -32 11 -31 11 -29 10 -27 11 -26 11 -23 11 -21 10 -21 11 -18 11 -18 10 -17 11 -16 11 -16 11 -15 10 -15 11 -15 11 -14 11 -14 10 -14 11 -14 11 -14 10 -13 11 -13 11 -13 11 -12 10 -11 11 -10 11 -10 10 -9 953 1456 100 MP stroke 11 -10 11 -8 11 -176 10 -374 910 2024 5 MP stroke SO 10 0 910 4615 2 MP stroke gr 3009 4936 mt (the learning steps) s 573 3298 mt -90 rotate (the maximum eigenvalue) s 90 rotate 3043 696 mt ( the original Hessian) s 3043 1287 mt ( the constrained Hessian) s 3043 3568 mt ( the EM-equivalent Hessian) s gs 0 0 560 420 MR c np end eplot epage end showpage %%EndDocument endTexFig 435 1772 a Fw(\(a\))959 b(\(b\))0 1883 y(Figure)15 b(3:)k(The)14 b(largest)g(eigen)o(v)m(alues)j(of)d(the)g(matrices)h Fo(H)q(;)8 b(E)1078 1866 y Fn(T)1104 1883 y Fo(H)t(E)s Fw(,)k(and)j Fo(E)1333 1866 y Fn(T)1360 1883 y Fo(P)6 b(H)t(E)16 b Fw(plotted)f(as)f(a)g(function)h(of)0 1951 y(the)g(n)o(um)o(b)q(er)h(of)e(iterations.)20 b(The)c(plot)f(in)h (\(a\))e(is)i(for)e(the)h(exp)q(erimen)o(t)i(in)f(Fig.)j(1;)c(\(b\))f (is)i(for)f(the)g(exp)q(erimen)o(t)0 2020 y(rep)q(orted)g(in)h(Fig.)k (2.)952 2817 y(19)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF