(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: kernelICA_icassp.dvi %%CreationDate: Fri Feb 21 11:19:32 2003 %%Pages: 4 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Bold Times-Italic Times-Roman Courier CMMI9 CMR9 %%+ CMR6 CMMI6 CMSY6 CMSY9 MSBM10 CMR5 CMSY5 CMEX10 CMBX9 Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE -t letter kernelICA_icassp %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.02.21:1113 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]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/IEn 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 IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/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 A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A 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/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 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 A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 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/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/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: 8r.enc % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "1 July 1998", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "Encoding for TrueType or Type 1 fonts % to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both % ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def 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$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$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 %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 81 /producttext put dup 101 /tildewide put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 25 /approxequal put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 68 /D put dup 70 /F put dup 75 /K put dup 82 /R put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D0037D33F24E2FAC2B0009AD3C8350CDF8CC 65BCA87979C36D14CB552E9A985E48BE4E88ECA16DF418749AF04FDD2B0E1380 D281BB2476BB45FF30946B247DFD7F57305FA87E50CA338121C71CDFDF927A9C 77FF14CB4A1D6D80356FB1171ED38C37702350497B44E42CE31DB2F493807DAA 15B887C671199A54C4C1294BC520F5538C15556BC43C9F62342B121C6DCD6C5F 491DA47FF360201EE21C08A781ED0589A6DF91B99FE118B9B29E4F068672E52F 1A06C514D91C4C937D4E642503392B1CD1BDE8C0299A06A62B1CA35D52CE191C E158AE14C8461EA0BBACF0753810DAD78CC310D3B03036A51783E4859A85A30C 4C3A88FA9A89B80AD09FB386DBA4EEBD778053B9AFA642C06A6D449D26CD1E0C 1C8574070B25BFFCF644B0E779B4BB02FF022179B35156DDF8376FA56BB095F8 D69D6C743B5F683EEBC9093BA2CD4EC2F93F21A3E2D6B4E1FCEE4FDFEFE4DA7C E62EDFF3B17B8776077BC1E65DE1CE535C78269ABA119C6765B4A38EE55B082D 71C76089D9F3F7D8B266B4342817B9E4DA1E47D6D92D88F8FF0D748029072182 AA4E00D302B27F633ACB10989DFAB2E9A5122EE59D59D6DE42465472DA9684A0 655BA778C120184D3EF04737C71B97579858E07264E1B778CE3B673997603B6C CEAF0AD69AE462C4140FDA452E1CCBAA9910072F6A5EB442B9F6854EC836AD24 E94F7C2F1830233BAB1E70CAE6EB4C4F7612E1AEC771FD4601F23F98069AEED3 1C7E57D6B8B068CC0E30CD9E73215BA00011E92D859835DAD0144906B3ECFF3A 96A3A00AAC9F81FB0EE5DF8910028879DC8D0DF6C2FC45C1695AA1F180BA85AA 1A62696CFBFFB947DA5B515E812553E35CCF588754B81D24BEABB15749DCA059 AEE9436F0AF97F9ABBD3177A63E816EA5F182D9B5C6B50955298B8CE6E75C294 5D103F4A79094ABE291CEEF398D023FED8BB248DBB1AC9E90436756F72FD4FA0 E7ED58CD32266B6E743AA0E5A3966271FBEF4B5E770B0854678C1F8BDD92238E 256D317022A9F3261A143DE84A0E42765F5D05A0F23C4FF6176DBCDF4575F13A 5FA8B62E00C2F5D5667B2FE8B2E796242BFC02E34836151B0CEAF8E727241842 71B9354D557929DA5307EC6498A17C3B23FF577427A8EECC6A0C60989D8E5262 CBEA85B302B886405DFC638754DAC7A93E03D3AC6F906AE4286C21B44CC72854 E100E5FA12E3762F1B4B07F19A184957C1380C22CB8EF070CB6C15FE7AC5FD25 D001257868871DE4F98287C43E88BA7D0B3A16565C6743E7390B6A26B79ECD02 3331FB40CA90E2BC540974C247EBDD54FDD00DB28839ECC8A960EC62F6A8FD52 8762BAF094A323A5614625BB89ABED3B7F9ADD966FB21029DABB523A414F86E4 531C15559B62CDF905C759DA0187B748BA88F9AE2E84CCE5060FC64C277179D2 E6A92162D43A7E0724580E4F7FF1FC1A0D5B4F6D37E7E35F302958CBC535AC8A 852F0B294503E3C6F56275147B4203F1EE9D68A543796717CD044EB639A3D8A1 15863CD629A79AEF3F9687058C718BED3D15F2AF2420D5D130EFD6416BC46C86 BD64FD6D16E0C46CC0E00BD65CA096D5FF8694298F8BF63A900FCE96FBADDE2A E6BB7BFA7C5E9E3D257252CD759AD86A6EF1D94E0A554A0BCA2DD3F9DABC365D 76DB1FA9346C8A1E485FA49A8D50751D385D7FCD3E31BA28F619C5E2158A9A11 CAFEAB098EF69A19BF2088DAF600BE2232E8C29A76FBF3865EF1BD9B6A10B32A 46A697D08D13A78D3B08524894DF749B89BC869DB5CB74D3B0CDC5220B471314 E0DFECA56EF5A9A09F37AA7C10520AF58085BFE4AE891B2D9C12F3DEF9FCBE84 4563466764A3BE573D3FE76415A415399109106E42A5DCD579CDD2458AFEA17E 9A58422F446BFA60DB4D1BAC84BE102BD5EDE2B88F2190889EC717A6016DD1D3 A3756CD15AC9CC43694AFA11456C045C8C8D504BF2418EB2EA47861840DDDFF7 37D16716BFCCEC0311A67853206F340020EEE342BCE90B9C8CE397E2B6DEBB7E D90661F43FE2D42DBB473E99C8C8A3A3E6F5047A87238A5612B140117F241503 EEF7FCAF7B1D9EFEF7B28D06615A381BA275F459DCD929A97F307D9506E1A87B 663A525C211CF0E14EB4EB1192F195659794AD3A8C6C87D35BDFF62FC11B7FC2 DABBDE051ECF3356634B59478ABB986A54A1486EBC0DA8138AC48C31A78FB5E3 43790DB937EEF78E216ADFEF3CF776A572479875BFE81B04F2EA7DAC6EBE02AD 58ADD26E0913999B2CD7DE3D378A206E6AD0D32212FD545F5D400ECFD4FE1E86 8FF0D667DEF632454AC094EB634E36655BAA88469F0124E1024978F98F6186D6 92B415A09C02FF88440618CC72140C5F600140A7A1D616655568CE9039EBC21F 56684CAEBEB40889A3A16CEF461631464E893DDF9082943B36B0F8F8EC4F4FA1 4262B9A29A7F0B3066764CD99CCAA5BB0C113FAFF99F9EFFF25D87BFDCB4C35B 4F48011FFE72DC8595F414A1DEBC5DCF49D5D0BF58C48DEBB0596895A73E12EF 2661BE04C9F15F5FCCB419C3D3D505225A432C820024663456154096E7313942 585CB6D3166A478A47C48FD09BD5EB53AF8BB1F6BD082AF6A1577FF310BAC317 9489E5D479E4BB37BFC732ACD5C4A0CA4751C0DDDAD0C10EE36EC711EE77F851 8C6F69FEB9D50F678CEABF0ED9AF766583AE68CC131942B3F56D87973D13FB55 1EA13CBE15C1978BB8ABDF419D7E7550ABECB3A7750BFCE1C8ED535A8E190755 50CDD96DC201BB2D70F076D21FEF24306E32C055A7BF005F1ED23B82531DBAEA 24ADD26A0F609023FDEC9AE21C3257CDB9E297F58A442493517A99CB4B89479D 25CDF47A33BA7D85ADC69CA96E44E7D85215E7C688F820CC6DE9E00458573149 F9C67234FC87807C66CFF8B7C134CCED3BBB26E75C07BADF7527B253C8D6038B EC561C2AEA8D77F25DD0CB683D6CBDC44846D246B5529E25FBD4E9F3BBC9E7D2 E4164E9E06F527D7BCA8D4FA1E03F2F6527A4F2A6819F993C6AD9985C61A08CA 5CCC76341D75096E2DC9AE531E7909BA31BF0665DE0DE5E0AE2F7606822B7FDE D553C0098008F3F35191F9F49E6183219CEF906D0356954DBF3C85419407A86B 7862F0A0DB23E7971B3595197F2FB17F14193DF8CBAA8FF2406F5EE314379D63 4152AE408C5C13E8A3585AF4C5324F1DD36DEBE5409A9C4D73F0AB35D7D4A606 C7F7589CCA402084C7DD9E5FEC32D67121DF1B030A6C7B4D8FAD3495129278B3 93CF3B6E8A2849937EE99FF417879FD3B59147F72F3DA25ECC989E31D8982D12 D4D42E355050EC81FF189427EDE4F19CAD750DD8FE6ED251931838FB45A8027B 011E1F8AD6A14D0F20A6AB5CA9FA2F280CA22F296379DACF91A511DDB3066633 8FC3FF50F21B7FA579F8E2B78DFDD19ED1C887F69209A3A40C36EAB21AB3B45D 2469EEC163ADA1BDE27735D98A88597AC40C80B491C45873EBC1CEBB4330E497 45F1E61ED3080FDB0DDD914166A6DA8A708DCF08BF1B715D93B08A77219C022C AEC65F6469C875995B68B995E8AD2E8DF2FB3335EE25AB907E075142F56E2A35 34AF9EF350A9AF4C420D0B05B4266666B26493E4C5EB961E1148885B3AA77631 FAECDE405DD5DDD6F6891B6035EBED70E0B90F06FCE5AF16BDD489E0CAF31DCC E9072503ADD862E50A62BB2875938F9150B0F171A0581704782FFAFFEAC8F51D CD4F71267A9B8E562AB3206B8A7356E48979CFFCF13CE9A084ADB7FA4EC951DC 558A8DC14E2F907E55C28A556408D9C3C61E3C425F3FC47817816DE0DCC15E9F EC7B28F7C503F0E3F7D37F29B2EE9AD54BA1B68BCF566395B0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 50 /element put dup 62 /latticetop put dup 70 /F put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 20 /kappa put dup 24 /xi put dup 26 /rho put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 78 /N put dup 97 /a put dup 98 /b put dup 102 /f put dup 105 /i put dup 106 /j put dup 109 /m put dup 120 /x put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 8 /Phi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 61 /equal put dup 94 /circumflex put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 108 /l put dup 109 /m put dup 111 /o put dup 112 /p put dup 114 /r put dup 116 /t put dup 118 /v put dup 120 /x put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 85 /U put dup 87 /W put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 109 /m put dup 114 /r put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D6F5CA5FED4F9AC9ADE41E04F9FC52E758C9F45A92BED935 86F9CFDB57732045913A6422AD4206418610C81D882EE493DE9523CC1BFE1505 DD1390B19BC1947A01B93BC668BE9B2A0E69A968554239B88C00AF9FBDF09CCD 67D3B2094C11A04762FE8CC1E91D020A28B3C122D24BEAACF82313F4604F2FEF 6E176D730A879BE45DD0D4996EF0247AEB1CA0AB08FF374D99F06D47B36F9554 FAD9A2D3CE451B7791C3709D8A1DDDEFBD840C1B42AB824D5A0DFF0E0F15B0B7 22AEEB877FF489581DA6FA8DA64944555101EB16F7AB0B717E148B7B98D8DBFD 730C52937E226545CF8DC3E07C5BA30739BAFCD0F2B44275A6D503F582C0FB4F 449963D0AD2FAFDE33BA3D77BCA9D1DF878DDAFCA2E22CC4BACD542B282164C7 97C2BDE318AF9D501CA21F6E662E7AAB75A5F24D2C182E598D175D44E88AB19A E7CD59584F95B389183EE21B525BF52A3F23C0FE5383A5565A19361D716F508C AAB78411CA5A4D27552CC1C435760D5A89D535B71C593E755C616661363308DA A683F54ED0C23FB2C225A008392B0B719F66F11A946A090B7C00B662A3C69599 B4ECB0CC70C85C4BBBF207E0026F6C7A19F2ACFB7A60804FC98A4BFFD7BFFF2B 9529E6D9D4238002BBC255BC62959D6F3381FE06E0621B879D5FE5B541D45A1E 759A6E7DC32B1D1632368D09A97039DF255B6492B1B2B7E2C1434E8306ECA7D3 5A79B6D614B4979F10988BC76ED53A5F45315CD7DA216221F842FD0F3E050DD2 BAC23C984D506D8F7D614BCB6B244F5F41321549BB0BD041FBF3053307168680 3435E9C9457C62AA6772AB002A095370E43DECFC47CC4339FB4341AF2E2A5740 C1C88BF1CE6302C0FDCBEDF1FACF1B6759F698882B24A7CC477C60EF661BE461 09B428CF27479BB6E50DA9FBC126D66A05E7EB88C0AABE9BFB56FF0630E3A7B6 84C31AD7576EF009D62D8B754E1779B99FFD8AB1C8542590217C6173E164F646 BE728A819D30AAC81B62F0AB2A4F83662AA33DB2CAC94A3032F8E9F870ED7733 E5B2D67E18F23604232FCCB12F900CD57C3DA61FBC8811E0C69374B376DEF4E3 EE9FF186E7134954DD7ED0149CB25D351599D748134171B72D2E0FD15B8B42DD 45FD7C5C2422536B608DF7D51C25030158AD09345240570E400268DA37467807 D81E647BA4BF69DBCEAE4422A7E5B8247F82E0AE5684F5402A9DE6E1A5FBC993 767E25022439D56B9F54C56AEDEA3EC26A2FC2D5D4DD894F981DFD71F0F22E48 AC55159148CBA0A26EFF699CF747958EF685CC99EB778494E437E580C881E43E 919E25759CEBABD1970F18E35B57DF8B81DB7FB10A9BA2BBF59A04DF5866B462 008313D2E4C9F45E70E1252CEFE88DA538789C92A451DBFFD0F95DA79D949651 441356928A64879BA3A003B0C178F172735E996B1728A80A54DD61262C14A2D1 058D5EC265D3B038BC5F20B0935944FE5BFD9AA25139C74E24896A83C3EABCF4 47A573DE7D802F5AEAD7FEBD7D3A941AD82A6F0F5109320EB11ABB64DB5F725A 37D124B1D433C3B3D746D174943C045CE677959D714CB48E23773398EAF15A93 1B90C034BFBB14610E3281B4D3769CCE764370449C30CC0756DB45A911A761EA 9D66B28BB15E22B859C73B29ACB277DFAE1224CD2C820B79CD40939D7AD1447F F6EFA83FDD342C1604A3BFF7FA7030215CD03AF81A7F86120CDE53A85B4DC61E 8B2E23A0173F414E316A89F46B7BA411EBD6D6BB6605573B5E8CF469923EF747 BBF9231BB333C28B21E11C8B924C29CB83F66CBCDAE9D22D4C76C7F90214DD64 F9A232A3754B6D0CA43852AE204896A5E874D57E8A340F070DFA4A3F0C214374 032CE784AE8D611AEFC2EBA5B7CD6B43721D652B67305E14AFB8DB1801500CA2 CC7C20B5DFCDF5F43B336DE68D667EA59323E1E097393F5B74AAAC68C63CFDA5 8E95A399DFBDEE972198E5189667F5C0D2F02AB2A37FB7BCCF9788C6D91ED21C 1C946EB05F870E3F06B3E0AD5435779A981BA621261463ED874162D804D62FAF 800C4664FA71BA32B671147266BB1825C84B6FEF6EE7DD072384EE91A1D16C74 64EB073B1A2D2FCAC7E7A3A9454A87F0BD573CCB1AEEBBBD3FA593CA01BCF528 D68E27CAC1DA4715EDE9236A33770306A85FE2EE9F81F3477EE316BB3575CB25 9CE0F59197B180664D1821F8CBFA66B41A5DEE7E8E052E761161CDE0EB05D0F2 8701AA5234BE41D15E208DF88A51E9E6DFCC38D82C0232678BB0DEDF046E6F48 FE69F12803F679B1747A943C9235DD012624EA7CEA2FF6B84E6B8ACF90F65B9F 85D071FD1ABC7044A1F6C994D9E756E24C2C2A98C68A40F0D417E1AB032A06B7 3C703DA940E3D478C140ACBFE154F4F710F043CAAB6B9197B219751E3F30611B 4F96596A146D7336C262E310C7B592D74739079C98FC30D8FE2B438E51205AC3 7310802A29837CE1B98832ACC4C41736A550E3BE1FE7A2B9BE0FDA2F14715162 94966480B0AABDB35251E623688134323CAF15B322E52DE833E476C90560A996 49CA00B72B4691EC3256364C670CD34EB07C12B1ADBB1B0CE8445C8BE773C1E4 32510D7B7030931F3089305DECFA35DE13E968C524002923C46A662C4EB1BF70 EB7F872CC3F57FBEAAB7064A7596BF9AD8566E2959881B4EB0169D4141387CF6 97FF168A700D24276C22843D9EB1F9674DAA1D50F15540A6CA9B9D9E1868619E BD137B9C8E03650F10DB01CD0BD6B66D62DB2AB3EB54F71D51E4787BF071AB45 71D3803A48F43E109D41890AC3AA00C8707E9B172EC372ED074C43ED9CAF121C FDE2D0725F3165E1598A52E6832B4D59C17F1A5B17D35F2CFA7F69CD97913F5D B49EF0E7ADA4C0CEEDD003A29E8EA3AE6974B9459BC09F449E36C67A5EBBDEDA ACF5EA597ADF0921DC25230515DBBAAB53377202EBA87F4578DDF5132FD6483A 5665EF21F022FCB85EBDD7F417090AEE5D6E4503119C0A389E9FE9F1D2097A92 ED664B8138DBE26C5AA77B518675FD009E2914398D0BD93D2FB95577EE5FA8EA 076E07B32E02B4AE09ACD47018EDFC036F82039C10359DE9D2BFF35D1A11DFE6 19246FD8C6D59AC25EB03BB034429D0748184FCBB9E1E3568299A3834729AB28 1CD4AB4833AF4D1E073D275A24B6962C372DF710315B8BFC1866C5232B287ADB 76FCC20E33EBEB6C1E7B1A58C85D5B6F425269E8FD9B176495093B75896EF15F F6EDAD995F1EA1729F0F73E8B8B7438362368EB25D91A23358BA877944F50011 FFB0B713837CDFDC8316892309004433A2B97A3CD3745B4D9336EC3BCF790321 2D9DA3B25857FC0395A34B0DE4AE1E49DDF3CCAE6F1D53C33E08E4DB49A251F0 DBC54BE2C9691A0D5BC9864B79FDFCAFC1C881488861F1B9C53E5DDB34FBE6EA 68AD00D516E3ADFDDE3D955A4B476BB32C211FC9D79C5F4C3691E1EC20763530 E5D6F41A7CC2DAD8B20A9B1EC70F6E4ED9434A81CAD33522BB652BF70C0A92D6 E33DAF3CF765B94706035DFCDAB193F86FD995239410006382987032C2AACCFF 2E3EB501B31B6154E661F5BFD19D5E4C2620694FCCF95B03B561CB96C62809EB CFEC6656A8B5D815751820D2E72AE3FC2404CDDA30373639AC3C648F052C8B82 E4BB88979DB843F284991F377E6BB7B1F590A0FF5BEA7B47E3D7CFD7FC15C84B FF3738440B80C388319608490CB67287533354084857520EE3F5DB55D930E43A AFD7136647D875ACE6BD4294EB8C33990ECD9E0FF07855F59C7F72AD52F4BD2C 1CB192382AEF8D0118FC1B725F1CB52060747F05F4589673F46727CA648133A2 291CEEF398F29C91E8B37751AB5FF13CE3210FF0003648820679C4E763E09ED6 CDA39C0A7BCBCC53E24D7791A712B8B1D16DB8487687B377B90860D697002C8D 6D0B2364EC96EC6F8CCB4DFBAAE1A2CE0715D755D7B094823032441B380307C2 D6CF0FE56523B8CFB39ED7AD9096F906186A6CB1B3AD2131430AD1F65FA8A327 E1043DF475E2AA7AA971D27EECE9DE380063E6864BC831A5B7404DE98F5310E7 B7C01196FC61282693FFD2D567ABB272F4DA30E45F843ACCA29E732C8C490635 0E12504F35C9584C7AAD4670FEE8A01556CB313FE0ABC833A21B416239A129E3 99D454ED36A411DF4F5F0D3F426ED4868AB57EFB2B66BF7CE14AF592BE8834B5 965A3DB993CB8340243F08857305F64CCBE97B56680F50A886CBC2714684B1FD F947EB160CFA52D09B9D8805A7A1B30527663B4A447F2DF9888477B10FC93884 A3BE3F18D87F2D773604D0BB8A3AF8E3356D570868A5A8BD756D2F62F4C68D44 877710C2605F4D1B24B9E7A598D1825AAA064EADA80A04F2A6716EDDEE7BFF5F 5DE6CCF733AF8139A76107F567A0C5BBC296EB8ACEA8A1747F5F30EE8C37B523 95DE1DE4F9CC08108ABC4FF91942154F3F2B6A0D4B1AD9AFD907714AE5D55091 CDEB6DB23899C37F5E922174EB61868060368DEFC52634B989ABA89CC6310794 16F9CA01E1A7C51685EEB1377878F7969FF2CD727CBCBC51F5F8EF27656E9836 6C7FC3DA99A5E77D79DB5A84A6F600A150906FE280EBAD5C8667B7C0DBDC809B E0905CE0A2712AE0EEE5C30FDF767B43AE19118F222BE8B51A9998E4C43DE812 A85559265C23F42E2AA270BF70E933DF1A03D81711D6FEF3768090A9ACBC5180 8AAE36B96B9AE5B446A09A918C18A14F8725F494577FFD0DF9F49A81BE18A93B EE095C3253A7F5209532974AE235F4D81B5EE74435140EA26095B97811BC706C 4B02221202F7A55A3A4BFD8D80CDE98615EBD8F84A674FABA5964A1B6D0E1AC5 E930EDAB43DD177FEE11F2EB1AF8ABBCEC3CAE456CF3E45E7A2164166A3687D7 E8E01E74A7AD5C468996F58762E118428BAE1FD6C52CD120D4B27EA9C19C0403 18100DD2450871D7DA0D45417D162CC24B857BFF9960F9EE1FEBCBEC3C156045 F7C9DC2C386489B3E2276E4C8FEBC919F99AEE7FBC696B1643F04FD5F5DC9810 24CB82880E4B958E6F9AF6978DB219BFB88977642F6DE185E6AD0C2A9C923A78 BCC31FFEDEFE772A93FB009A6C0083F442280E129CBDDEBC128581654890D621 058500EB3A266A3A81819889FD2E1C30C3FACE325558A2298927015E2AAFBBC0 6A27AD95B4218D07CBAEA04B7C6F2CB80423FDFE0227A6FCC7909FFB7594E292 06D140DFEE400B317528912ACE2A1664ECF8E21569E1562AA90FF96DB4D6A3E4 6A91F028A77A42F2F2DFC28BFA6D70B24786C3B3631492409CE130568ED65B9A 762F03B77270B9FC07F0C8B55DEC77EF861F04A321DD37448C77B5AE7AF4A30C 479E1F1A2278263C27D8F4197D731FCE90260BECA9A78D5CD1269CA96B95F526 11399188133C1B50CDF0C50B8B5A0CFE8F664A9C244597FD6A90C20E70A50205 FA43363DFD666A574A85C12D47C5155F8C4F88AB3F91E1D3143AF467C2C7795A E509D681EDCB77FD4BE38BC2FBB0C95D423D78C9D6C395A979CBABFE4990F888 42B0A7E67F53909A43CC5F5D130EB53CF0F8904807A6662DD3F25D187395A887 63BD20CF97BBE8E04951A922A939A6891C95F7CA5722D89FF428009CCCDF5045 3D7F3E6A9A6642760850991C340AEE1C2A36504B3B6DE4937CF53A3B5FBAA9BF 7F304AB50DC49D6D010F8BB0A07ACB1443C29BBFAF23E09C247EF8103CA4C575 E350F1E4B3ACE03FD994975D8DC6EC47B027AA3EE0A1B7F5700A72A76CD122EF E8AF6A1E3F6BD7A5932A8DBF5B79886F08087C74F5C6846BD89F295FC9D267E6 6DB0F8858081981075F4B270269ACE4100A729455CC007E99DADABABC3EEC832 FC4D314B6A4EA9EDE6C62902B1A8F65A5B17328377132CDC6808BE2334F398DD 743F946DBFD14D51F2534923698B4478122258F414A7F740DD5F0F55F8B26736 D7C864894E573E5F150C84DC69957F4F50F19301D8D8D256943BFCF1452AB22F F82F2235978EE061D5BC2B7BB823B44E55E579AFF6AB28332AA9272AD4A666EA 74B88847C93F58CEBB21233D9DD17CE73D76307CF27AC9123B4B01DF0864BE23 60F9EF62333302D9BFB3D95157103EB24D368F81C407B0186E1DFA4CD57390A0 6DDBA4696232429062842A2C5955C2DAA7076CE5FFE1F7A2B576EF2C9D9B8B9E E5E407FACE8AC2CE636C6CA8D0B554C752E4AB983460A2901177E8B4FFD51D58 E42CC490D5A4E3E8464AC6326B6404EE6EB049700479995C619845CA47F4FD27 6787077FA69D796F8817D2B437BF5AB2E2AA69C2FEB82ABFE2E5339147F6BA08 61366816D25649B65F32B2EDA9AD8127F0F4B9DA30A0E4E42DB5AC98940B7694 130485AD33F6EEBC2BC2C789E87909DB52082D42027FB256FCBBDC5C699E3C92 494F06D5CBEA607C355E1F37525C64EBD3DFCE61DA3AEAEDC0E82AB1BFEB543A 42B13DE7F88576DFB6D627EDC78B5B0D930F035E8B23DF7A8CEF64ABFFC99915 C5182420BE0C4604406ED7A6716DB20900D1D9CC7E37DFFF60C3DB555F4DEB9E 2488DB60DBC386C74BC5B31D1B02D1C40508C8160E1F8763F89AA95AD1090AB2 42A240A81AC267362798DBC33366619C4E4274504630DE1F67D965F8AD189841 D03B4A81764C16419D01E379AB48EA519D3457586216911E1141275217807159 084F639B11DED8F8D958EC0A87F0CD62D16354AC2B5B6DE58C36B298FE83F197 5E5B9F124D47E8A8394420C17FBE9A20EEBDA394B189236EC0E49EAE66B38846 92F0F9E897A4F0484448AB663938322585D7393C3C71516DA66169D7B6560026 D5E9E53E3077443C33B97B0EBAA5B0DF2A2DD8C1CFDF6A3570095C877276626A C4D2A8AB0222E548168C32AC7BA5AFB31D9D0018080D45E91614305218D420CE 3E717DBE7566D308768136BD146C1C9560B8EE459C654BB53CEF583287C95D0F AD1C6A193A6388D4802CF93D923669B8C8E5B0B2AB4887AC8DFAE94A9F396222 3306DEEC86D43397B25460C23C30ED34FEDD9FB44C8E3A489820B8066652C01A E7C41201F780963575F09ADEC8483EB6C6826EE6E45E755762231F053A7A1BD0 7029783076419970AC0762687382B75430487C2DFFE2E34A41F94271B746A481 46399671B432F33BC10C0A5A032A3EA856E8FE85F78E992AF9DB5652E44A0176 BFBEB3F41FDC0959DCE15B46DB43B8F4EC3D15DB4285E7D8472B0494DF6332F2 C77A6DED11916871F42A119CABB8E9A558702E6FBF940D521BB979B66F255E27 6E8D5BBD375C54240AF623E454F86639F91B776A98DD13BFD8644D0D7859BBD7 36F59F8906E2653D26B8C12D4FA4F11A99FD3CB441F80661DBD1F7B7B5A08FB8 369EF5D88E975058C33621B22FA012754EDFB954FD649BA19A035E6E09165653 60DE990D142733AF26F4276A15452F8432B530D464E06BAC297DC019E0C53792 11DFB732D73BF9A4E1C25AC1A95501614B03FDF1CDA9AEB8F43BD14B611B5540 D3B7C8B349927E367C8D062C0C3A47E90F2224DD93CFD2438723D43A3CC050BE E51BC06946BA0FFD564D5998E62911B34456223C533EDA9AF795F5D8241BE0D6 12035E28762340F7A8C97B0A7BF1EBB1E4CE44AD132DBDD2C095C20DAF0CB1F7 9C183A392BD8D79CAD7ADEE658A0C732A38A601C47BEC17E4040E9364F3D68B9 490ABC0362A75B871242796FF41EC0C10EFFEB837C22FD3BFE404C98C13BF11D 75705DAB30B77E133DE34ABA4C38FA155F10BAFAA78E6E6089872719F24ABBDB 676D09919FAC581409ABE68B9A98AC60663F7108781D84C06F547976D3AC4F32 FBA7EE80CD9D1122EC83182B12C87E18AAE5283F495D7594DEE070F1AF6E0B45 8325DFAAE75BBB0F7FE51ED46B968016D95333CB37FC2A917685AF1F95E95D0E 1537D3E045F1E243F66C4EFBEF070601A09C781FB0C0D7AAF6813EF98A4D6F10 E68C7EC7D2490E5874DB49E59374E1ABAF0850983BD0E692008E8F63573AA947 6F7ED497E249655A9955B10803FD0DF21EA6DD5C843A778C01C7A6646555A2E6 E2114910ECECD06D08329CEFA7ECF7564F769CD5F56E911591145A07D420352B CF5C5C668110228935481BE16C3FAAA9B441969F503FAB6FFD607E2FC09792DA 133F8954BAB474CC909D3DD2710644B8DCF76E40998525640A2121C93762C40F BF80D9AB0AAC34C18E444C28C467EBB69D1D1CBA876C0218BD76570365538284 B1E613219C184C7DDA4731BC3E686F8E7305DD9D75E080141F76EC5372450091 702DF9A97882B48849C9CB99B488A5AA8FF62452A6008CAE15C5F2D0EDBD24C1 DE4B59A1B98E12C0C185E9ED85A149D5B5D6E046E2A81885F3AF9866F9C0D5D1 E2DD780AA66C8042321E239792EA8B008665C13AD88DEEECD9FA81310FC80977 6E2D3C87078166554A45904341E4F10C8042DCCD96EA43C9B063723C015A891E 75D90A11312F988EEAE65D397E123FA1129A0816E087788D1A368FDF0DC2A713 F4C36B0BB5D6C42A6D9C4B5E315573FE373F5C97E4522CEED0A9C8485067E502 15E90E6FBBA686A80C65A829B0AB336CD40FD4987A9B3C787A4D2E2B0B5FCBA1 6F9ECFC41DDA5E3EFA79D8041E30586E6C39351408D939B43E397ACE6A50452F 2BA7050ECDF0513C75A7181497D2823C32ED3662DE5D4B2351A6526DE061FA0C 0615AA6D0BCAA4E4C9A549610C7BF6E1F3E25B8635B264A60863987ABF87E381 B933991CB7D40B3D5770352C8E06AAD26A1AEAC298AFA0169E1F3403D46E7327 A9C96EE0932AAC894A9B71E3B419E6D2A335A928F4E645C7649E9AB52EB77EB4 0D89CEB4EAD8ED32B35CF16A4BF1D990C19AE7AA78A2A66439D999B4C110E475 2EAC53C51A16B86B690F850D5EB5147D41C0F16013B80BEE2A5075E7C84E0ACA 594F5F0B1E5FC8BB3FE6791C174C645CA18A0F94284C6E5720373FB61B37B3C9 EDD6A9EDDCA09DEEDE72B6F6DDCF787B422E6F9916828F7D9595E47BA5B4801D 3037E4ADF88E193485060F8ADFF8FE58ABC9FEF8BCB7DB165641A87F4D5D48DD B259D0BC795AC3B367D4EEF61EAF35D63A2618468FF4A47B1AE9A5BCD2ED520F AB20E280E74F6558D96813192C80C60C3E4BFC1FD6C7781700E0473AD80AB125 802F6FD77A45D03AC6C12FF3FFA4E3EB6A69E1F1BB835921965DBD6AF25EB882 D7D24B98C320BBEC64C9A1F2280B44C62C30898B0CF0E08CBEF22632FC86A4FF 4450BB18FED617C99AFC6E9570BAD89729F0ECBA9D893BFE0C05D12D4F75850F C695E40C0BA2A1D92C84795605876366C227C33C5196CC17E017C18DB03015B5 126E79271B2ED1B95137A40030F3F3C333CF71048EDE1E24F6DC8869B60DE757 7C31D3D591EEF700436B328E5C4A2FF666C9319129AD3F9D06D6558AD964EDA7 202AFC7BCEF32361166A4671E3232E99BA1C2043CDCB28128037F7FF37632E99 6EDBC3C137DDDA1C3CE62FC04B289042ED7FBC9D10B7E53233380B8CA1B2C525 1CC0CCECFDE33882CC6A997C82893F5CDCA4743C41559DF1F9D209EB6D23167D 93DA90EBF0BA9BA472E002628E300ED34ABBF14DE7223E6A4CEBC38401807A75 E1333E99643B40058F1E6949466D038CA899F7538386300EEFD4C65E2AC64290 99CFFDF4173C6E3BC64B74915B94B32004DBCBD4D1A5CDFB12DCD9942441A9FD ABE00F013CA9BF08D41A7B89358DB9DE97794E903751746B9D7262DA2BBD456B 63D4C426264220A3E6005D3839B7571AD01EF874A1301A5778D2D063392E2E65 6E79A3C77305678A6BC92EC16EEF7D289A37686E98BAF0BB17ADF4F74DBF745B F1F13160DFC2CF17106942501757D4DCB51FED9F8C6033BE206090BA95FCC461 39F9BEEBDDCF8888A6CA061B5CB23A8D913EE15AE9550FFA464DAD5BD5A39F34 A0E4DBD25100F916DE42AB42EB40CCAB0024821704201C9626E2FF9DBFDF89C1 4837B3690EE01E482CB877D7DE072D6CE3F3949D9EE7B838B5922FA94FB2B5B1 D18E60FF9DCC1DFA2A9850B1E051163D91C44F6B903E5ACE69226D1B0D6A6E9B 8E6899F86B310C9FAB9D1E97D68DE170187E944EFB29B1F51281C10359420507 739C857029FF91056A7CD34732843691108A2BF9DF9AAF9D86A7EF2CF50F6CC8 B4E75F34058A2F5E096842EE4F22236004CDE40E782F067D09D4C9D72C2C4453 4F1A790E881E95274BB03711F5B49133129BA9CA9D11FCC5E99E0D6C8DEEBAA4 31522763E40A78EB457F95418DA01994456E54C659EBB07717562D1FBA6F0235 70AF17F413C4659A263FC34A763256F5C3394C0F67671A168B5E49FECE2CBE9D CE4295DD6C419DFA4ECAF1592C8F24F3F21E07B281D1B8BAF503202E3238D665 4A80681464C97C5355D28A0464CF49B48E6E32B7ABBD286FE3F078FFA1DF583C 6B911B6944ADD39B9CC3462684B33560E519146952A439B1B5C16777E936ED10 F5B625704690C4DEA71C973D9161A38E852E342CBD96A420D7D3E1A2217B87F1 DF9275DC81761C1E7731933DF3293CFECCEAC0C307B31E2B4BCBD6D3C3EE8A70 EDD372696E4D9FA1D8E2B2E0F1E46D0BE36E82A7F7980BA3AAFFEEF0C8B2DCE4 64237878FCA5EA9A01C5BB0228C5B3C7D4B89FA840F8B5FE5686B1D9A7BABD07 C749B7C9B1D367A657E478707C2FB3F316A23BA1EB14E4ABBF716F3C65AFAF25 5EC5A5BBCE0E43740410378294B7D2D79E2929AAAB0B2E7A5D234977CBB48354 BAFEFD18C7337024F4B898CCAD0591BD1FC5191321F5CED389BB7295354B6A48 8F41A354A3F5A0F494C235D7787A4147664379CAFA0B16AAD0852C4E8190EED9 EF4AA135B5FE64F4A80D0A1B9E5EA05B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 40258431 52099146 1000 600 600 (kernelICA_icassp.dvi) @start /Fa 139[18 26 5[48 18 6[29 3[33 97[{ TeXBase1Encoding ReEncodeFont }6 66.4176 /Times-Italic rf /Fb 133[29 33 33 48 33 33 18 26 22 33 33 33 33 52 18 2[18 33 33 22 29 33 29 33 29 3[22 1[22 7[37 9[22 5[44 1[48 13[33 4[17 22 17 41[37 2[{ TeXBase1Encoding ReEncodeFont }35 66.4176 /Times-Roman rf /Fc 206[25 49[{ TeXBase1Encoding ReEncodeFont } 1 49.8132 /Times-Roman rf /Fd 206[44 49[{}1 74.7198 /CMBX9 rf /Fe 154[43 19[73 61[57 57 16[35 35{}6 74.7198 /CMEX10 rf /Ff 185[41 70[{}1 41.511 /CMSY5 rf /Fg 205[28 28 49[{}2 41.511 /CMR5 rf /Fh 173[54 82[{}1 74.7198 /MSBM10 rf /Fi 150[30 30 38 38 19[65 6[58 4[55 1[59 12[0 0 3[51 16[77 7[60 22[60 1[60{}15 74.7198 /CMSY9 rf /Fj 185[44 7[48 11[42 49[48{}4 49.8132 /CMSY6 rf /Fk 135[35 10[54 2[25 22 2[30 3[27 34 18[48 16[32 1[19 25[39 6[32 1[28 3[36 5[28 14[{}15 49.8132 /CMMI6 rf /Fl 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Fm 135[41 1[41 1[30 1[30 1[43 38 1[64 21 4[38 1[34 43 34 1[38 2[38 32[60 8[38 1[38 38 38 4[60 1[30 30 31[55 4[53 3[{}24 74.7198 /CMR9 rf /Fn 133[36 38 44 5[35 4[68 2[31 26 2[37 36 13[73 1[52 4[49 58 61 74 52 65 1[34 1[60 2[63 55 1[58 3[38 1[21 21 24[48 5[44 40 1[34 2[45 44 5[34 2[49 11[{}33 74.7198 /CMMI9 rf /Fo 173[40 3[43 1[37 6[37 69[{ TeXBase1Encoding ReEncodeFont } 4 59.7758 /Times-Roman rf /Fp 133[29 33 1[50 33 37 21 29 29 1[37 37 37 54 21 33 1[21 37 37 21 33 37 33 1[37 10[46 54 1[37 46 1[46 1[50 62 42 50 33 25 1[54 1[46 1[50 1[46 8[37 9[19 25 19 2[25 25 37[37 2[{ TeXBase1Encoding ReEncodeFont } 44 74.7198 /Times-Italic rf /Fq 87[25 16[75 37 1[33 33 24[33 37 37 54 37 37 21 29 25 37 37 37 37 58 21 37 21 21 37 37 25 33 37 33 37 33 3[25 1[25 1[54 1[71 54 54 46 42 50 1[42 54 1[66 46 54 29 25 54 54 42 46 1[50 50 54 5[21 21 37 37 37 37 37 37 37 37 37 37 1[19 25 19 2[25 25 1[58 34[42 42 2[{ TeXBase1Encoding ReEncodeFont }74 74.7198 /Times-Roman rf /Fr 133[33 37 1[54 37 42 25 29 33 1[42 37 42 62 21 2[21 42 37 25 33 42 33 42 37 7[54 1[75 1[54 50 42 54 1[46 58 54 71 50 58 1[29 58 58 46 50 54 54 50 54 7[37 37 37 37 37 37 37 37 37 37 1[19 25 45[{ TeXBase1Encoding ReEncodeFont }55 74.7198 /Times-Bold rf /Fs 134[60 3[60 1[60 60 2[60 60 1[60 60 60 1[60 1[60 60 60 60 60 60 32[60 17[60 46[{ TeXBase1Encoding ReEncodeFont }18 99.6264 /Courier rf /Ft 134[50 2[50 50 28 39 33 1[50 50 50 78 28 50 1[28 2[33 44 50 44 1[44 11[72 1[55 14[72 66 66 72 7[50 1[50 2[50 1[50 1[50 3[25 44[{ TeXBase1Encoding ReEncodeFont }30 99.6264 /Times-Roman rf /Fu 140[39 39 2[50 50 1[28 2[28 50 2[44 50 44 1[50 14[61 4[83 2[44 33 2[61 3[61 19[25 46[{ TeXBase1Encoding ReEncodeFont } 18 99.6264 /Times-Italic rf /Fv 166[72 4[66 55 72 1[61 78 72 94 66 78 1[39 3[66 72 72 1[72 65[{ TeXBase1Encoding ReEncodeFont } 15 99.6264 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: Letter letter %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 695 260 a Fv(KERNEL)26 b(INDEPENDENT)g(COMPONENT) e(AN)n(AL)-9 b(YSIS)590 501 y Fu(F)k(r)o(ancis)23 b(R.)i(Bac)o(h)374 722 y Ft(Computer)f(Science)i(Di)n(vision)447 832 y(Uni)n(v)o(ersity)c (of)j(California)392 943 y(Berk)o(ele)o(y)-6 b(,)25 b(CA)g(94720,)f (USA)296 1053 y Fs(fbach@cs.berkeley.edu)2467 501 y Fu(Mic)o(hael)f(I.) i(J)n(or)l(dan)2259 722 y Ft(Computer)g(Science)g(Di)n(vision)2243 832 y(and)f(Department)h(of)g(Statistics)2332 943 y(Uni)n(v)o(ersity)d (of)j(California)2278 1053 y(Berk)o(ele)o(y)-6 b(,)24 b(CA)h(94720,)f(USA)2151 1163 y Fs(jordan@cs.berkeley.edu)599 1475 y Fr(ABSTRA)l(CT)-186 1602 y Fq(W)-6 b(e)18 b(present)h(a)g(class) f(of)h(algorithms)g(for)f(independent)j(component)f(anal-)-186 1689 y(ysis)26 b(\(ICA\))f(which)h(use)h(contrast)f(functions)g(based)h (on)g(canonical)g(cor)o(-)-186 1776 y(relations)22 b(in)g(a)f (reproducing)j(k)o(ernel)f(Hilbert)e(space.)32 b(On)22 b(the)g(one)h(hand,)-186 1862 y(we)28 b(sho)n(w)g(that)f(our)h (contrast)g(functions)h(are)f(related)f(to)h(mutual)g(infor)o(-)-186 1949 y(mation)19 b(and)g(ha)o(v)o(e)f(desirable)h(mathematical)g (properties)g(as)f(measures)h(of)-186 2036 y(statistical)j(dependence.) 38 b(On)23 b(the)g(other)g(hand,)i(b)o(uilding)e(on)h(recent)f(de-)-186 2123 y(v)o(elopments)g(in)f(k)o(ernel)h(methods,)g(we)f(sho)n(w)h(that) f(these)g(criteria)f(can)i(be)-186 2209 y(computed)29 b(ef)n(\002ciently)-5 b(.)48 b(Minimizing)28 b(these)f(criteria)g (leads)g(to)g(\003e)o(xible)-186 2296 y(and)d(rob)o(ust)f(algorithms)g (for)g(ICA.)f(W)-6 b(e)23 b(illustrate)f(with)h(simulations)g(in-)-186 2383 y(v)o(olving)28 b(a)g(wide)g(v)n(ariety)g(of)g(source)h(distrib)o (utions,)g(sho)n(wing)g(that)f(our)-186 2470 y(algorithms)19 b(outperform)h(man)o(y)g(of)f(the)g(presently)g(kno)n(wn)h(algorithms.) 452 2673 y Fr(1.)45 b(INTR)n(ODUCTION)-186 2823 y Fq(Recent)15 b(research)g(on)g(k)o(ernel)g(methods)g(has)g(yielded)g(important)g(ne) n(w)g(com-)-186 2909 y(putational)26 b(tools)f(for)f(solving)i(lar)o (ge-scale,)g(nonparametric)h(classi\002ca-)-186 2996 y(tion)e(and)g(re)o(gression)g(problems)h([10].)40 b(While)24 b(some)h(forays)g(ha)o(v)o(e)g(also)-186 3083 y(been)k(made)f(into)f (unsupervised)j(learning,)g(there)e(is)f(still)g(much)h(une)o(x-)-186 3170 y(plored)22 b(terrain)f(in)g(problems)h(in)m(v)o(olving)h(lar)o (ge)e(collections)g(of)h(mutually)-186 3256 y(interacting)37 b(v)n(ariables,)k(problems)c(in)g(which)g(Mark)o(o)o(vian)i(or)17 b(general)-186 3343 y(graphical)24 b(models)f(ha)o(v)o(e)h(e)o (xcelled.)36 b(These)23 b(latter)f(models)i(in)f(f)o(act)g(ha)o(v)o(e) -186 3430 y(se)n(v)o(eral)16 b(limitations)g(that)g(in)m(vite)g(k)o (ernel-based)h(initiati)n(v)o(es;)g(in)e(particular)m(,)-186 3517 y(the)o(y)24 b(are)f(almost)h(entirely)g(based)g(on)g(strong)g (parametric)g(assumptions,)-186 3603 y(and)c(lack)f(the)g (nonparametric)h(\003e)o(xibility)e(of)h(the)g(k)o(ernel)h(approaches.) -61 3690 y Fp(Independent)32 b(component)h(analysis)e(\(ICA\))e Fq([8)q(])h(is)g(an)h(interesting)-186 3777 y(unsupervised)e(learning)e (problem)h(in)e(which)h(to)g(e)o(xplore)g(these)g(issues.)-186 3864 y(On)15 b(the)g(one)h(hand,)h(ICA)d(is)h(hea)o(vily)g(based)i(on)e (structural)g(assumptions\227)-186 3950 y(vie)n(wed)k(as)g(a)f (graphical)i(model)f(it)f(is)g(a)h(directed)g(bipartite)f(graph)i (linking)-186 4037 y(a)j(set)h(of)f(\223source)i(nodes\224)f(to)g(a)f (set)g(of)h(\223observ)n(ation)h(nodes,)-5 b(\224)25 b(in)f(which)-186 4124 y(the)17 b(lack)f(of)h(edges)g(between)h(the)e (source)i(nodes)f(encodes)h(an)f(assumption)-186 4211 y(of)24 b(mutual)g(independence.)42 b(On)24 b(the)g(other)g(hand,)i (the)e(ICA)g(problem)h(is)-186 4297 y(also)h(strongly)g (nonparametric\227the)h(distrib)o(ution)f(of)g(the)g(source)g(v)n(ari-) -186 4384 y(ables)e(is)g(left)f(unspeci\002ed.)40 b(This)24 b(is)f(dif)n(\002cult)h(to)g(accommodate)h(within)-186 4471 y(the)d(\(current\))g(graphical)g(model)g(formalism,)g(in)g(which) g(all)f(nodes)i(must)-186 4558 y(be)e(endo)n(wed)i(with)e(a)g (probability)h(distrib)o(ution.)29 b(It)21 b(is)g(here)g(that)g(we)g (will)-186 4645 y(\002nd)f(k)o(ernel)i(methods)f(to)g(be)f(useful.)28 b(W)-6 b(e)20 b(will)g(sho)n(w)h(ho)n(w)g(k)o(ernel)g(meth-)-186 4731 y(ods)26 b(can)h(be)f(used)h(to)f(de\002ne)g(a)g(\223contrast)g (function\224)h(that)f(can)g(be)g(used)-186 4818 y(to)f(estimate)g(the) h(parametric)f(part)g(of)h(the)f(ICA)g(model)g(\(the)h(source-to-)-186 4905 y(observ)n(ation)h(edges\),)h(despite)e(the)g(absence)h(of)f(a)g (speci\002c)f(distrib)o(ution)-186 4992 y(on)j(the)g(source)g(nodes.)50 b(As)28 b(we)f(will)g(see,)i(compared)g(to)f(current)g(ICA)-186 5078 y(algorithms,)i(the)e(ne)n(w)g(k)o(ernel-based)i(approach)f(is)f (notable)g(for)g(its)f(ro-)-186 5165 y(b)o(ustness.)-61 5252 y(W)-6 b(e)19 b(refer)i(to)f(our)h(ne)n(w)g(approach)h(to)e(ICA)g (as)g(\223)r(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)r(.)-5 b(\224)16 b(It)k(is)-186 5339 y(important)j(to)g(emphasize)h(at)f(the)g (outset)g(that)i(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)20 b(is)j(not)g(the)1975 1475 y(\223k)o(ernelization\224)30 b(of)e(an)g(e)o(xtant)g(ICA)g(algorithm.)50 b(Rather)m(,)30 b(it)e(is)f(a)h(ne)n(w)1975 1561 y(approach)22 b(to)e(ICA)g(based)h(on) f(no)o(v)o(el)h(k)o(ernel-based)h(measures)e(of)h(depen-)1975 1648 y(dence.)34 b(W)-6 b(e)22 b(introduce)h(tw)o(o)f(such)h(measures.) 34 b(In)22 b(Section)g(3,)h(we)f(de\002ne)1975 1735 y(a)29 b(k)o(ernel-based)i(contrast)f(function)g(in)f(terms)g(of)g(the)g (\002rst)f(eigen)m(v)n(alue)1975 1822 y(of)d(a)f(certain)h(generalized) g(eigen)m(v)o(ector)h(problem,)g(and)f(sho)n(w)g(ho)n(w)g(this)1975 1908 y(function)e(relates)e(to)h(probabilistic)g(independence.)33 b(In)22 b(Section)g(4.3,)g(we)1975 1995 y(introduce)30 b(an)e(alternati)n(v)o(e)h(k)o(ernel-based)g(contrast)g(function)g (based)g(on)1975 2082 y(the)f(entire)f(spectrum)h(of)g(the)f (generalized)h(eigen)m(v)o(ector)h(problem,)h(and)1975 2169 y(sho)n(w)20 b(ho)n(w)f(this)g(function)h(can)f(be)g(related)g(to) g(mutual)g(information.)2498 2373 y Fr(2.)45 b(B)n(A)l(CKGR)n(OUND)18 b(ON)h(ICA)1975 2528 y Fq(Independent)j(component)f(analysis)f(\(ICA\)) f(is)g(the)h(problem)g(of)g(reco)o(v)o(er)o(-)1975 2615 y(ing)g(a)f(latent)g(random)h(v)o(ector)g Fn(x)h Fm(=)h(\()p Fn(x)3000 2623 y Fl(1)3034 2615 y Fn(;)13 b(:)g(:)g(:)h(;)f(x)3249 2623 y Fk(m)3307 2615 y Fm(\))3337 2583 y Fj(>)3408 2615 y Fq(from)19 b(observ)n(ations)1975 2701 y(of)h Fn(m)g Fq(unkno)n(wn)i(linear)e(functions)h(of)e(that)h(v)o(ector)l(.)27 b(The)20 b(components)i(of)1975 2788 y Fn(x)h Fq(are)f(assumed)i(to)f (be)g(mutually)g(independent.)37 b(Thus,)24 b(an)f(observ)n(ation)1975 2875 y Fn(y)36 b Fm(=)d(\()p Fn(y)2210 2883 y Fl(1)2244 2875 y Fn(;)13 b(:)g(:)g(:)g(;)g(y)2452 2883 y Fk(m)2511 2875 y Fm(\))2541 2843 y Fj(>)2618 2875 y Fq(is)24 b(modeled)i(as)f Fn(y)36 b Fm(=)d Fn(Ax)p Fq(,)25 b(where)g Fn(x)f Fq(is)h(a)g(latent) 1975 2962 y(random)g(v)o(ector)f(with)g(independent)h(components,)i (and)d(where)g Fn(A)f Fq(is)g(an)1975 3049 y Fn(m)6 b Fi(\002)g Fn(m)21 b Fq(matrix)h(of)f(parameters.)32 b(Gi)n(v)o(en)22 b Fn(N)29 b Fq(independently)-5 b(,)24 b(identically)1975 3135 y(distrib)o(uted)f(observ)n(ations)i(of)e Fn(y)s Fq(,)g(we)f(hope)i(to)f(estimate)g Fn(A)f Fq(and)i(thereby)1975 3222 y(to)c(reco)o(v)o(er)g(the)f(latent)h(v)o(ector)g Fn(x)f Fq(corresponding)i(to)f(an)o(y)g(particular)f Fn(y)j Fq(by)1975 3309 y(solving)e(a)f(linear)g(system.)2100 3399 y(By)30 b(specifying)h(distrib)o(utions)f(for)g(the)g(components)i Fn(x)3633 3407 y Fk(i)3659 3399 y Fq(,)h(one)d(ob-)1975 3485 y(tains)19 b(a)g(parametric)h(model)f(that)g(can)h(be)f(estimated) h(via)f(maximum)h(lik)o(e-)1975 3572 y(lihood)i([5].)27 b(W)-6 b(orking)21 b(with)f Fn(W)34 b Fm(=)24 b Fn(A)3002 3540 y Fj(\000)p Fl(1)3105 3572 y Fq(as)c(the)h(parameterization,)g (one)1975 3659 y(readily)27 b(obtains)f(a)g(gradient)g(or)g(\002x)o (ed-point)g(algorithm)h(that)e(yields)h(an)1975 3754 y(estimate)2272 3735 y Fm(^)2250 3754 y Fn(W)35 b Fq(and)26 b(pro)o(vides)g(estimates)f(of)h(the)f(latent)g(components)j(via)1980 3849 y Fm(^)-43 b Fn(x)21 b Fm(=)2144 3830 y(^)2121 3849 y Fn(W)11 b(y)21 b Fq([8].)2100 3939 y(In)14 b(practical)h (applications,)h(ho)n(we)n(v)o(er)m(,)g(one)f(does)g(not)g(generally)g (kno)n(w)1975 4026 y(the)k(distrib)o(utions)g(of)f(the)h(components)h Fn(x)3089 4034 y Fk(i)3115 4026 y Fq(,)e(and)i(it)e(is)g(preferable)h (to)f(vie)n(w)1975 4113 y(the)24 b(ICA)g(model)g(as)g(a)g Fp(semipar)o(ametric)g(model)g Fq(in)g(which)g(the)g(distrib)o(u-)1975 4199 y(tions)i(of)g(the)g(components)i(of)e Fn(x)f Fq(are)h(left)f (unspeci\002ed)i([6)q(].)43 b(Maximiz-)1975 4286 y(ing)22 b(the)g(lik)o(elihood)h(in)e(the)h(semiparametric)g(ICA)f(model)h(is)f (essentially)1975 4373 y(equi)n(v)n(alent)c(to)d(minimizing)h(the)g (mutual)g(information)h(between)f(the)g(com-)1975 4468 y(ponents)27 b(of)e(the)g(estimate)k Fm(^)-43 b Fn(x)33 b Fm(=)2901 4449 y(^)2879 4468 y Fn(W)10 b(y)27 b Fq([7].)41 b(Thus)25 b(it)g(is)f(natural)i(to)e(vie)n(w)1975 4555 y(mutual)c(information)g(as)g(a)f Fp(contr)o(ast)h(function)h Fq(to)e(be)h(minimized)g(in)f(esti-)1975 4642 y(mating)h(the)f(ICA)f (model.)2100 4731 y(Unfortunately)-5 b(,)16 b(the)f(mutual)f (information)h(for)g(real-v)n(alued)g(v)n(ariables)1975 4818 y(is)g(dif)n(\002cult)g(to)f(approximate)j(and)e(optimize)h(on)f (the)g(basis)g(of)g(a)g(\002nite)g(sam-)1975 4905 y(ple,)k(and)g(much)g (research)g(on)f(ICA)g(has)h(focused)g(on)g(alternati)n(v)o(e)f (contrast)1975 4992 y(functions)24 b([8,)f(7,)f(1)q(].)34 b(These)23 b(ha)o(v)o(e)g(either)g(been)h(deri)n(v)o(ed)f(as)g(e)o (xpansion-)1975 5078 y(based)31 b(approximations)g(to)f(the)f(mutual)h (information,)j(or)d(ha)o(v)o(e)f(had)i(a)1975 5165 y(looser)23 b(relationship)g(to)f(the)g(mutual)g(information,)i(essentially)e (borro)n(w-)1975 5252 y(ing)27 b(its)f(k)o(e)o(y)h(property)g(of)f (being)i(equal)f(to)f(zero)h(if)e(and)i(only)g(if)f(the)g(ar)o(-)1975 5339 y(guments)d(to)e(the)h(function)h(are)e(independent.)33 b(In)22 b(this)f(paper)m(,)i(we)e(de\002ne)p eop end %%Page: 2 2 TeXDict begin 2 1 bop -186 83 a Fq(tw)o(o)26 b(no)o(v)o(el)h(contrast)f (functions.)46 b(Minimizing)26 b(them)h(will)e(lead)h(to)g(tw)o(o)-184 170 y(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)16 b(algorithms.)-118 356 y Fr(3.)45 b(MEASURING)17 b(ST)-7 b(A)g(TISTICAL)17 b(DEPENDENCE)h(WITH)622 443 y(KERNELS)-186 592 y Fq(In)e(this)f (section,)h(we)g(de\002ne)g(the)f Fi(F)7 b Fq(-correlation,)17 b(a)e(measure)i(of)e(statistical)-186 679 y(dependence)24 b(among)e(random)g(v)n(ariables)g Fn(x)994 687 y Fl(1)1028 679 y Fn(;)13 b(:)h(:)f(:)g(;)g(x)1243 687 y Fk(m)1301 679 y Fq(.)30 b(F)o(or)21 b(simplicity)-5 b(,)-186 766 y(we)21 b(restrict)g(ourselv)o(es)h(initially)f(to)g(the)h(case)g(of)f (tw)o(o)h(real)f(random)h(v)n(ari-)-186 853 y(ables,)h Fn(x)53 861 y Fl(1)108 853 y Fq(and)g Fn(x)282 861 y Fl(2)316 853 y Fq(,)f(treating)g(the)g(general)g(case)g(of)g Fn(m)f Fq(v)n(ariables)i(in)f(Sec-)-186 939 y(tion)d(3.4.)25 b(\(It)18 b(is)h(also)h(w)o(orth)f(noting)h(that)f(the)h(restriction)f (to)g(real)g(random)-186 1026 y(v)n(ariables)i(is)f(again)h(for)f (simplicity;)g(a)h(similar)e(measure)i(of)g(dependence)-186 1113 y(can)i(be)f(de\002ned)h(for)g(an)o(y)f(type)h(of)f(data)h(for)f (which)h(Mercer)g(k)o(ernels)g(can)-186 1200 y(be)c(de\002ned\).)-61 1286 y(W)-6 b(e)32 b(assume)h(that)g(we)f(are)h(gi)n(v)o(en)h(a)e (reproducing-k)o(ernel)k(Hilbert)-186 1373 y(space)15 b(\(RKHS\))f Fi(F)22 b Fq(on)15 b Fh(R)p Fq(,)g(with)f(k)o(ernel)h Fn(K)5 b Fm(\()p Fn(x;)13 b(y)s Fm(\))h Fq(and)h(feature)g(map)g Fm(\010\()p Fn(x)p Fm(\))p Fq(.)-186 1460 y(In)53 b(this)g(paper)m(,)k (our)c(focus)26 b(is)f(the)53 b(Gaussian)26 b(k)o(ernel,)56 b Fn(K)5 b Fm(\()p Fn(x;)13 b(y)s Fm(\))33 b(=)-186 1547 y(exp)o(\()p Fi(\000)p Fm(\()p Fn(x)p Fi(\000)p Fn(y)s Fm(\))226 1515 y Fl(2)259 1547 y Fn(=)p Fm(2)p Fn(\033)382 1515 y Fl(2)417 1547 y Fm(\))p Fq(,)15 b(which)f(corresponds)j(to)d(an) h(in\002nite-dimensional)-186 1634 y(RKHS)j(of)h(smooth)h(functions)f ([10)q(].)-186 1826 y Fr(3.1.)45 b(The)18 b Fi(F)7 b Fr(-corr)o(elation)-186 1964 y Fq(Gi)n(v)o(en)26 b(an)g(RKHS)e Fi(F)7 b Fq(,)27 b(we)f(de\002ne)g(the)f Fi(F)7 b Fq(-correlation)27 b(as)e(the)h(maximal)-186 2050 y(correlation)15 b(between)g(the)g (random)g(v)n(ariables)g Fn(f)1082 2058 y Fl(1)1117 2050 y Fm(\()p Fn(x)1191 2058 y Fl(1)1225 2050 y Fm(\))f Fq(and)h Fn(f)1428 2058 y Fl(2)1463 2050 y Fm(\()p Fn(x)1537 2058 y Fl(2)1571 2050 y Fm(\))p Fq(,)g(where)-186 2137 y Fn(f)-149 2145 y Fl(1)-95 2137 y Fq(and)k Fn(f)68 2145 y Fl(2)122 2137 y Fq(range)g(o)o(v)o(er)h Fi(F)7 b Fq(:)13 2279 y Fn(\032)53 2287 y Fj(F)189 2279 y Fm(=)128 b(max)332 2326 y Fk(f)362 2336 y Fg(1)394 2326 y Fk(;f)443 2336 y Fg(2)475 2326 y Fj(2F)578 2279 y Fm(corr)q(\()p Fn(f)778 2287 y Fl(1)813 2279 y Fm(\()p Fn(x)887 2287 y Fl(1)921 2279 y Fm(\))p Fn(;)13 b(f)1022 2287 y Fl(2)1057 2279 y Fm(\()p Fn(x)1131 2287 y Fl(2)1165 2279 y Fm(\)\))474 b Fq(\(1\))189 2482 y Fm(=)128 b(max)332 2529 y Fk(f)362 2539 y Fg(1)394 2529 y Fk(;f)443 2539 y Fg(2)475 2529 y Fj(2F)759 2433 y Fm(co)n(v)q(\()p Fn(f)938 2441 y Fl(1)973 2433 y Fm(\()p Fn(x)1047 2441 y Fl(1)1081 2433 y Fm(\))p Fn(;)13 b(f)1182 2441 y Fl(2)1217 2433 y Fm(\()p Fn(x)1291 2441 y Fl(2)1325 2433 y Fm(\)\))p 588 2465 968 4 v 588 2533 a(\(v)l(ar)g Fn(f)773 2541 y Fl(1)808 2533 y Fm(\()p Fn(x)882 2541 y Fl(1)916 2533 y Fm(\)\))976 2511 y Fl(1)p Fk(=)p Fl(2)1072 2533 y Fm(\(v)l(ar)f Fn(f)1256 2541 y Fl(2)1291 2533 y Fm(\()p Fn(x)1365 2541 y Fl(2)1399 2533 y Fm(\)\))1459 2511 y Fl(1)p Fk(=)p Fl(2)1566 2482 y Fn(:)112 b Fq(\(2\))-61 2666 y(Clearly)-5 b(,)26 b(if)e(the)h(v)n (ariables)g Fn(x)721 2674 y Fl(1)780 2666 y Fq(and)h Fn(x)957 2674 y Fl(2)1016 2666 y Fq(are)f(independent,)j(then)d(the) -186 2753 y Fi(F)7 b Fq(-correlation)31 b(is)e(equal)i(to)f(zero.)56 b(Moreo)o(v)o(er)m(,)34 b(if)29 b(the)h(set)g Fi(F)37 b Fq(is)29 b(lar)o(ge)-186 2840 y(enough,)e(the)d(con)m(v)o(erse)i(is)d (also)i(true.)38 b(F)o(or)24 b(e)o(xample,)i(it)e(is)f(well)h(kno)n(wn) -186 2926 y(that)19 b(if)f Fi(F)27 b Fq(contains)19 b(the)g(F)o(ourier) g(basis)g(\(all)g(functions)g(of)h(the)f(form)g Fn(x)i Fi(7!)-186 3013 y Fn(e)-150 2981 y Fk(i!)r(x)-20 3013 y Fq(where)29 b Fn(!)42 b Fi(2)e Fh(R)p Fq(\),)31 b(then)e Fn(\032)698 3021 y Fj(F)790 3013 y Fm(=)40 b(0)29 b Fq(implies)f(that)h Fn(x)1390 3021 y Fl(1)1452 3013 y Fq(and)h Fn(x)1633 3021 y Fl(2)1695 3013 y Fq(are)-186 3100 y(independent.)f(In)20 b([3)q(],)g(we)g(sho)n(w)h(that)f(the)g(con)m(v)o(erse)i(is)d(also)i (true)f(for)g(the)-186 3187 y(reproducing)h(k)o(ernel)e(Hilbert)g (spaces)g(based)h(on)g(Gaussian)f(k)o(ernels.)-61 3273 y(F)o(or)j(reasons)h(that)f(will)g(become)i(clear)e(in)h(Section)f (4.3,)i(it)e(is)g(useful)-186 3360 y(to)k(w)o(ork)h(on)g(a)g (logarithmic)f(scale;)k(in)c(particular)m(,)i(we)f(de\002ne)f(our)h (\002rst)-186 3447 y(contrast)21 b(function)g(as)f Fn(I)456 3455 y Fk(\032)488 3466 y Ff(F)565 3447 y Fm(=)j Fi(\000)718 3416 y Fl(1)p 718 3430 31 4 v 718 3472 a(2)771 3447 y Fm(log)r(\(1)c Fi(\000)e Fn(\032)1074 3455 y Fj(F)1127 3447 y Fm(\))p Fq(.)27 b(Our)20 b(con)m(v)o(erse)h(result)-186 3534 y(implies)h(that)f Fn(I)223 3542 y Fk(\032)255 3553 y Ff(F)330 3534 y Fq(is)h(a)f(v)n(alid)i(contrast)f(function;)i(a)e (function)g(that)g(is)g(al-)-186 3620 y(w)o(ays)e(nonne)o(gati)n(v)o(e) h(and)g(equal)f(to)g(zero)g(if)f(and)h(only)g(if)f(the)h(v)n(ariables)g Fn(x)1752 3628 y Fl(1)-186 3707 y Fq(and)g Fn(x)-15 3715 y Fl(2)37 3707 y Fq(are)f(independent.)-61 3794 y(The)f(ability)g(to)g (restrict)f(the)i(maximization)g(in)f(Eq.)f(\(1\))i(to)f(an)g(RKHS)-186 3881 y(has)30 b(an)g(important)f(computational)i(consequence.)57 b(In)29 b(particular)m(,)j(we)-186 3967 y(can)26 b(e)o(xploit)g(the)g Fp(r)m(epr)m(oducing)i(pr)m(operty)p Fq(,)g Fn(f)8 b Fm(\()p Fn(x)p Fm(\))35 b(=)f Fi(h)p Fm(\010\()p Fn(x)p Fm(\))p Fn(;)13 b(f)8 b Fi(i)p Fq(,)27 b(to)f(ob-)-186 4054 y(tain)20 b(an)g(interpretation)g(of)g Fn(\032)580 4062 y Fj(F)652 4054 y Fq(in)g(terms)g(of)g(linear)g(projections.)26 b(Indeed,)-186 4141 y(the)46 b(reproducing)h(property)g(implies)e(that) 22 b Fm(corr\()p Fn(f)1254 4149 y Fl(1)1289 4141 y Fm(\()p Fn(x)1363 4149 y Fl(1)1397 4141 y Fm(\))p Fn(;)13 b(f)1498 4149 y Fl(2)1533 4141 y Fm(\()p Fn(x)1607 4149 y Fl(2)1641 4141 y Fm(\)\))26 b(=)-186 4228 y(corr)13 b(\()p Fi(h)p Fm(\010\()p Fn(x)148 4236 y Fl(1)183 4228 y Fm(\))p Fn(;)g(f)284 4236 y Fl(1)318 4228 y Fi(i)p Fn(;)g Fi(h)p Fm(\010\()p Fn(x)541 4236 y Fl(2)576 4228 y Fm(\))p Fn(;)g(f)677 4236 y Fl(2)712 4228 y Fi(i)p Fm(\))f Fn(:)18 b Fq(Consequently)-5 b(,)20 b(the)e Fi(F)7 b Fq(-correlation)-186 4314 y(is)21 b(the)h(maximal)g(possible)h(correlation)f(between)g(one-dimensional)i (lin-)-186 4401 y(ear)c(projections)g(of)f Fm(\010\()p Fn(x)486 4409 y Fl(1)521 4401 y Fm(\))g Fq(and)h Fm(\010\()p Fn(x)826 4409 y Fl(2)861 4401 y Fm(\))p Fq(.)k(This)19 b(is)g(e)o(xactly)h(the)g(de\002nition)-186 4488 y(of)25 b(the)h(\002rst)e Fp(canonical)j(corr)m(elation)f Fq([2])f(between)h Fm(\010\()p Fn(x)1352 4496 y Fl(1)1387 4488 y Fm(\))f Fq(and)g Fm(\010\()p Fn(x)1703 4496 y Fl(2)1738 4488 y Fm(\))p Fq(.)-186 4575 y(This)19 b(interpretation)i(will)d(enable)j (us)f(to)g(deri)n(v)o(e)g(a)g(computationally)h(ef)n(\002-)-186 4661 y(cient)e(algorithm.)-186 4854 y Fr(3.2.)45 b(Canonical)18 b(corr)o(elation)i(analysis)-186 4992 y Fq(Canonical)27 b(correlation)f(analysis)g(\(CCA\))f(is)g(a)h(multi)n(v)n(ariate)g (statistical)-186 5078 y(technique)21 b(similar)d(in)h(spirit)g(to)g (principal)h(component)h(analysis)f(\(PCA\).)-186 5165 y(While)e(PCA)f(w)o(orks)i(with)f(a)g(single)h(random)g(v)o(ector)g (and)g(maximizes)g(the)-186 5252 y(v)n(ariance)h(of)g(projections)g(of) f(the)h(data,)f(CCA)g(w)o(orks)h(with)f(a)g(pair)h(of)f(ran-)-186 5339 y(dom)26 b(v)o(ectors)g(\(or)g(in)f(general)i(with)e(a)h(set)f(of) h Fn(m)f Fq(random)i(v)o(ectors\))f(and)1975 83 y(maximizes)k (correlation)f(between)g(sets)g(of)f(projections.)53 b(While)29 b(PCA)1975 170 y(leads)16 b(to)f(an)g(eigen)m(v)o(ector)h (problem,)g(CCA)f(leads)g(to)g(a)g(generalized)h(eigen-)1975 257 y(v)o(ector)30 b(problem.)54 b(More)29 b(precisely)-5 b(,)32 b(gi)n(v)o(en)e(tw)o(o)f(random)h(v)o(ectors,)i Fn(x)3914 265 y Fl(1)1975 343 y Fq(and)25 b Fn(x)2151 351 y Fl(2)2185 343 y Fq(,)g(the)f(\002rst)f(canonical)i(correlation)g (between)f Fn(x)3458 351 y Fl(1)3516 343 y Fq(and)h Fn(x)3692 351 y Fl(2)3750 343 y Fq(can)f(be)1975 430 y(de\002ned)18 b(as)f(the)g(maximum)h(possible)g(correlation)f(between)h(the)f(tw)o(o) g(pro-)1975 517 y(jections)j Fn(\030)2264 485 y Fj(>)2261 530 y Fl(1)2316 517 y Fn(x)2360 525 y Fl(1)2412 517 y Fq(and)g Fn(\030)2576 485 y Fj(>)2573 530 y Fl(2)2628 517 y Fn(x)2672 525 y Fl(2)2725 517 y Fq(of)f Fn(x)2850 525 y Fl(1)2902 517 y Fq(and)h Fn(x)3073 525 y Fl(2)3107 517 y Fq(:)2155 649 y Fn(\032)p Fm(\()p Fn(x)2269 657 y Fl(1)2302 649 y Fn(;)13 b(x)2380 657 y Fl(2)2414 649 y Fm(\))83 b(=)g(max)2671 696 y Fk(\030)2699 706 y Fg(1)2732 696 y Fk(;\030)2779 706 y Fg(2)2826 649 y Fm(corr\()p Fn(\030)3025 613 y Fj(>)3022 662 y Fl(1)3077 649 y Fn(x)3121 657 y Fl(1)3155 649 y Fn(;)14 b(\030)3227 613 y Fj(>)3224 662 y Fl(2)3279 649 y Fn(x)3323 657 y Fl(2)3357 649 y Fm(\))474 b Fq(\(3\))2527 862 y Fm(=)83 b(max)2671 909 y Fk(\030)2699 919 y Fg(1)2732 909 y Fk(;\030)2779 919 y Fg(2)3148 814 y Fn(\030)3185 782 y Fj(>)3182 827 y Fl(1)3237 814 y Fn(C)3292 822 y Fl(12)3357 814 y Fn(\030)3391 822 y Fl(2)p 2836 845 902 4 v 2836 874 a Fe(\000)2871 935 y Fn(\030)2908 909 y Fj(>)2905 954 y Fl(1)2960 935 y Fn(C)3015 943 y Fl(11)3080 935 y Fn(\030)3114 943 y Fl(1)3148 874 y Fe(\001)3183 892 y Fl(1)p Fk(=)p Fl(2)3293 874 y Fe(\000)3328 935 y Fn(\030)3365 909 y Fj(>)3362 954 y Fl(2)3417 935 y Fn(C)3472 943 y Fl(22)3537 935 y Fn(\030)3571 943 y Fl(2)3605 874 y Fe(\001)3640 892 y Fl(1)p Fk(=)p Fl(2)3747 862 y Fn(;)93 b Fq(\(4\))1975 1071 y(where)24 b Fn(C)2236 1079 y Fk(ij)2314 1071 y Fq(denotes)h(the)e(co)o(v)n(ariance)i(matrix)f Fm(co)n(v)q(\()p Fn(x)3429 1079 y Fk(i)3455 1071 y Fn(;)13 b(x)3533 1079 y Fk(j)3565 1071 y Fm(\))p Fq(.)37 b(By)23 b(taking)1975 1157 y(deri)n(v)n(ati)n(v)o(es)c(with)f(respect)g(to)g Fn(\030)2804 1165 y Fl(1)2856 1157 y Fq(and)h Fn(\030)3016 1165 y Fl(2)3050 1157 y Fq(,)f(this)g(problem)g(is)g(easily)g(seen)h (to)1975 1244 y(reduce)h(to)f(the)g(follo)n(wing)g(generalized)h(eigen) m(v)n(alue)h(problem)f([2]:)2225 1317 y Fe(\022)2325 1378 y Fm(0)124 b Fn(C)2542 1386 y Fl(12)2284 1464 y Fn(C)2339 1472 y Fl(21)2528 1464 y Fm(0)2610 1317 y Fe(\023)13 b(\022)2739 1378 y Fn(\030)2773 1386 y Fl(1)2739 1464 y Fn(\030)2773 1472 y Fl(2)2811 1317 y Fe(\023)2889 1422 y Fm(=)21 b Fn(\032)3022 1317 y Fe(\022)3082 1378 y Fn(C)3137 1386 y Fl(11)3325 1378 y Fm(0)3122 1464 y(0)124 b Fn(C)3339 1472 y Fl(22)3407 1317 y Fe(\023)13 b(\022)3536 1378 y Fn(\030)3570 1386 y Fl(1)3536 1464 y Fn(\030)3570 1472 y Fl(2)3608 1317 y Fe(\023)3677 1422 y Fn(:)163 b Fq(\(5\))1975 1598 y(W)-6 b(e)18 b(need)g(to)f(be)h(able)g(to)g(solv)o(e)g(this)f (problem)h(in)g(feature)g(space,)g(and)g(thus)1975 1685 y(we)h(need)h(to)f(consider)h(a)e(\223k)o(ernelized\224)j(v)o(ersion)e (of)g(CCA.)1975 1876 y Fr(3.3.)45 b(Estimating)19 b(the)f Fi(F)7 b Fr(-corr)o(elation)1975 2013 y Fq(Let)23 b Fi(f)p Fn(x)2180 1981 y Fl(1)2180 2026 y(1)2215 2013 y Fn(;)13 b(:)g(:)g(:)g(;)g(x)2429 1981 y Fk(N)2429 2026 y Fl(1)2487 2013 y Fi(g)23 b Fq(and)i Fi(f)p Fn(x)2762 1981 y Fl(1)2762 2026 y(2)2796 2013 y Fn(;)13 b(:)g(:)g(:)h(;)f(x)3011 1981 y Fk(N)3011 2026 y Fl(2)3068 2013 y Fi(g)24 b Fq(denote)g(sets)f (of)h Fn(N)32 b Fq(empirical)1975 2100 y(observ)n(ations)25 b(of)e Fn(x)2503 2108 y Fl(1)2560 2100 y Fq(and)g Fn(x)2734 2108 y Fl(2)2769 2100 y Fq(.)34 b(The)23 b(observ)n(ations)i(generate)f Fp(Gr)o(am)f(ma-)1975 2186 y(trices)g Fn(L)2216 2194 y Fl(1)2273 2186 y Fq(and)h Fn(L)2456 2194 y Fl(2)2491 2186 y Fq(,)f(de\002ned)g(as)g Fm(\()p Fn(L)2942 2194 y Fk(i)2969 2186 y Fm(\))2999 2195 y Fk(ab)3092 2186 y Fm(=)29 b Fn(K)5 b Fm(\()p Fn(x)3325 2155 y Fk(a)3325 2200 y(i)3362 2186 y Fn(;)13 b(x)3440 2155 y Fk(b)3440 2200 y(i)3471 2186 y Fm(\))p Fq(.)35 b(The)22 b Fp(center)m(ed)1975 2273 y(Gr)o(am)27 b(matrices)g Fq([10)q(])f Fn(K)2675 2281 y Fl(1)2737 2273 y Fq(and)h Fn(K)2936 2281 y Fl(2)2997 2273 y Fq(are)g(de\002ned)h(as)f(the)f(Gram)h(matri-)1975 2360 y(ces)c(of)f(the)g(centered)h(\(in)f(feature)g(space\))h(data)f (points)h(and)g(are)f(equal)h(to)1975 2447 y Fn(K)2040 2455 y Fk(i)2088 2447 y Fm(=)e Fn(P)11 b(L)2281 2455 y Fk(i)2307 2447 y Fn(P)30 b Fq(where)19 b Fn(P)32 b Fm(=)21 b Fn(I)h Fi(\000)2903 2416 y Fl(1)p 2892 2430 54 4 v 2892 2472 a Fk(N)2955 2447 y Fd(1)d Fq(is)f(a)h(constant)h (singular)f(matrix)g(\()p Fd(1)1975 2533 y Fq(is)g(the)g Fn(N)12 b Fi(\002)t Fn(N)28 b Fq(matrix)19 b(composed)h(of)f(ones\).) 2100 2620 y(F)o(ollo)n(wing)27 b(the)g(spirit)f(of)h(the)g(deri)n(v)n (ation)g(of)g(k)o(ernel)h(PCA)e([10],)i(it)1975 2707 y(is)19 b(straightforw)o(ard)g(to)f(deri)n(v)o(e)i(a)e(\223k)o (ernelization\224)i(of)f(CCA,)e(which)i(turns)1975 2794 y(out)24 b(to)g(in)m(v)o(olv)o(e)g(substituting)g(products)h(of)f(Gram) f(matrices)h(for)g(the)f(co-)1975 2880 y(v)n(ariance)d(matrices)f(in)g (Eq.)f(\(3\),)h(and)g(maximizing)2759 3017 y Fn(\013)2808 2985 y Fj(>)2808 3030 y Fl(1)2860 3017 y Fn(K)2925 3025 y Fl(1)2960 3017 y Fn(K)3025 3025 y Fl(2)3059 3017 y Fn(\013)3108 3025 y Fl(2)p 2091 3048 1720 4 v 2091 3118 a Fm(\()p Fn(\013)2170 3092 y Fj(>)2170 3137 y Fl(1)2223 3118 y Fm(\()p Fn(K)2318 3126 y Fl(1)2369 3118 y Fm(+)e Fn(N)8 b(\024I)e(=)p Fm(2\))2705 3096 y Fl(2)2740 3118 y Fn(\013)2789 3126 y Fl(1)2824 3118 y Fm(\))2854 3096 y Fl(1)p Fk(=)p Fl(2)2951 3118 y Fm(\()p Fn(\013)3030 3092 y Fj(>)3030 3137 y Fl(2)3082 3118 y Fm(\()p Fn(K)3177 3126 y Fl(2)3229 3118 y Fm(+)17 b Fn(N)8 b(\024I)e(=)p Fm(2\))3565 3096 y Fl(2)3600 3118 y Fn(\013)3649 3126 y Fl(2)3684 3118 y Fm(\))3714 3096 y Fl(1)p Fk(=)p Fl(2)3820 3065 y Fn(;)1975 3241 y Fq(where)22 b Fn(\024)f Fq(is)f(a)h(small)g (positi)n(v)o(e)g(re)o(gularization)h(parameter)l(.)30 b(As)20 b(for)h(CCA)1975 3328 y(in.)30 b(Eq.)21 b(\(3\),)h(the)f (solution)h(is)f(obtained)h(by)g(solving)g(the)f(follo)n(wing)h(gen-) 1975 3415 y(eralized)e(eigen)m(v)n(alue)g(problem)g(\(cf.)j(Eq.)18 b(\(3\))h(and)g(\(5\)\):)2055 3489 y Fe(\022)2170 3549 y Fm(0)61 b Fn(K)2334 3557 y Fl(1)2369 3549 y Fn(K)2434 3557 y Fl(2)2102 3636 y Fn(K)2167 3644 y Fl(2)2202 3636 y Fn(K)2267 3644 y Fl(1)2356 3636 y Fm(0)2459 3489 y Fe(\023)-26 b(\022)2549 3549 y Fn(\013)2598 3557 y Fl(1)2549 3636 y Fn(\013)2598 3644 y Fl(2)2623 3489 y Fe(\023)2676 3594 y Fm(=)9 b Fn(\032)2785 3489 y Fe(\022)2843 3549 y Fm(\()p Fn(K)2938 3557 y Fl(1)2964 3549 y Fm(+)3025 3519 y Fk(N)c(\024)p 3025 3533 90 4 v 3055 3575 a Fl(2)3125 3549 y Fn(I)h Fm(\))3195 3518 y Fl(2)3370 3549 y Fm(0)3004 3640 y(0)142 b(\()p Fn(K)3279 3648 y Fl(2)3305 3640 y Fm(+)3366 3610 y Fk(N)5 b(\024)p 3366 3624 V 3396 3666 a Fl(2)3466 3640 y Fn(I)h Fm(\))3536 3608 y Fl(2)3560 3489 y Fe(\023)-26 b(\022)3650 3549 y Fn(\013)3699 3557 y Fl(1)3650 3636 y Fn(\013)3699 3644 y Fl(2)3724 3489 y Fe(\023)3861 3594 y Fq(\(6\))1975 3784 y(Since)21 b Fm(\()p Fn(K)2257 3792 y Fk(i)2302 3784 y Fm(+)e Fn(\024I)6 b Fm(\))2495 3752 y Fl(2)2550 3784 y Fq(is)21 b(necessarily)h(in)m(v)o (ertible,)f(classical)g(methods)h(can)1975 3871 y(be)d(in)m(v)o(ok)o (ed)h(to)e(solv)o(e)h(the)g(generalized)g(eigen)m(v)n(alue)h(problem)g (in)e(Eq.)g(\(6\).)1975 3968 y(Thus)d(k)o(ernel)g(CCA)f(reduces)h(to)g (\002nding)g(the)f(lar)o(gest)g(eigen)m(v)n(alue)i(of)3787 3949 y Fe(e)3775 3968 y Fi(K)3833 3976 y Fk(\024)3895 3968 y Fm(=)1975 3995 y Fe(\022)2231 4056 y Fm(0)203 b Fn(r)2507 4064 y Fk(\024)2546 4056 y Fm(\()p Fn(K)2641 4064 y Fl(1)2676 4056 y Fm(\))p Fn(r)2741 4064 y Fk(\024)2781 4056 y Fm(\()p Fn(K)2876 4064 y Fl(2)2910 4056 y Fm(\))2022 4143 y Fn(r)2057 4151 y Fk(\024)2097 4143 y Fm(\()p Fn(K)2192 4151 y Fl(2)2227 4143 y Fm(\))p Fn(r)2292 4151 y Fk(\024)2332 4143 y Fm(\()p Fn(K)2427 4151 y Fl(1)2461 4143 y Fm(\))f(0)2930 3995 y Fe(\023)2987 4101 y Fq(,)15 b(with)f Fn(r)3203 4109 y Fk(\024)3243 4101 y Fm(\()p Fn(K)3338 4109 y Fk(i)3364 4101 y Fm(\))21 b(=)g Fn(K)3561 4109 y Fk(i)3588 4101 y Fm(\()p Fn(K)3683 4109 y Fk(i)3709 4101 y Fm(+)p Fn(\024I)6 b Fm(\))3883 4069 y Fj(\000)p Fl(1)3965 4101 y Fq(.)1975 4336 y Fr(3.4.)45 b(Generalization)19 b(to)g(mor)o(e)g(than)f(tw)o(o)i (v)o(ariables)1975 4473 y Fq(It)28 b(is)g(straightforw)o(ard)h(to)f(e)o (xtend)h(CCA,)e(and)i(its)f(k)o(ernelized)h(counter)o(-)1975 4560 y(part,)g(to)e(the)g(case)g(of)g Fn(m)g Fq(v)n(ariables)h([3].)47 b(The)27 b(problem)h(becomes)g(that)1975 4646 y(of)h(\002nding)g(the)g (smallest)f(eigen)m(v)n(alue)i(of)f(the)f(generalized)i(eigen)m(v)n (alue)1975 4733 y(problem)20 b Fi(K)q Fn(\013)j Fm(=)e Fn(\025)p Fi(D)r Fn(\013)p Fq(,)e(where)g Fi(K)i Fq(is)d(de\002ned)i (by)g(blocks)g Fi(K)3601 4741 y Fk(ij)3678 4733 y Fm(=)h Fn(K)3824 4741 y Fk(i)3850 4733 y Fn(K)3915 4741 y Fk(j)1975 4820 y Fq(for)i Fn(i)30 b Fi(6)p Fm(=)f Fn(j)f Fq(and)c Fi(K)2478 4828 y Fk(ii)2556 4820 y Fm(=)29 b(\()p Fn(K)2740 4828 y Fk(i)2787 4820 y Fm(+)20 b Fn(\024I)6 b Fm(\))2981 4788 y Fl(2)3015 4820 y Fq(,)24 b(and)g Fi(D)h Fq(is)e(block)h (diagonal)g(with)1975 4907 y(blocks)e Fi(D)2250 4915 y Fk(ii)2323 4907 y Fm(=)h(\()p Fn(K)2501 4915 y Fk(i)2546 4907 y Fm(+)18 b Fn(\024I)6 b Fm(\))2738 4875 y Fl(2)2772 4907 y Fq(.)27 b(W)-6 b(e)20 b(still)f(refer)h(to)g(this)g(eigen)m(v)n (alue)i(as)f(the)1975 4993 y Fi(F)7 b Fq(-correlation.)2404 4962 y Fc(1)p 1975 5035 789 4 v 2066 5090 a(1)2095 5114 y Fb(See)22 b([3])f(for)g(a)g(detailed)k(e)o(xplanation)f(of)d(why)g (we)h(use)f(the)h Fa(smallest)i Fb(gener)o(-)1975 5189 y(alized)e(eigen)m(v)n(alue)h(in)d(our)g(general)h(de\002nition,)h(and) e(ho)n(w)f(this)h(accords)h(with)f(our)1975 5264 y(earlier)g (de\002nition.)k(In)17 b(brief,)h(the)g(de\002nitions)h(are)f(equi)n(v) n(alent)j(because)e(of)f(a)f(sym-)1975 5339 y(metry)h(property)h(of)e (the)g(eigen)m(v)n(alues)k(for)c(the)h(CCA)f(problem.)p eop end %%Page: 3 3 TeXDict begin 3 2 bop -61 83 a Fq(It)17 b(is)h(w)o(orth)h(noting)g (that)f(the)h(general)g(v)o(ersion)g(of)f(the)g Fi(F)7 b Fq(-correlation)-186 170 y(that)33 b(we)f(ha)o(v)o(e)i(de\002ned)f (does)h(not)f(characterize)g Fp(mutual)h Fq(dependence)-186 257 y(among)17 b Fn(m)f Fq(v)n(ariables,)h(b)o(ut)e(only)i (characterizes)g Fp(pairwise)f Fq(independence.)-186 343 y(Empirically)-5 b(,)19 b(this)g(does)h(not)f(appear)h(to)f(be)h(a) f(limitation)g(in)g(the)g(ICA)g(set-)-186 430 y(ting,)24 b(as)f(we)g(sho)n(w)h(in)f(Section)g(5.)36 b(Ho)n(we)n(v)o(er)m(,)25 b(in)e(situations)h(in)f(which)g(a)-186 517 y(measure)e(of)f(mutual)g (independence)j(is)c(required,)i(one)g(can)f(form)g(such)h(a)-186 604 y(measure)k(by)g(e)o(xploiting)g(the)g(general)g(f)o(act)g(that)f (mutual)h(independence)-186 690 y(can)16 b(be)g(e)o(xpressed)h(in)f (terms)g(of)g(pairwise)f(mutual)h(information)h(terms)e(in-)-186 777 y(v)o(olving)k(sets)g(of)g(v)n(ariables.)k(\(Thus,)c(for)g(e)o (xample,)g(in)g(the)f(three-v)n(ariable)-186 864 y(case)h(we)g(ha)o(v)o (e)g(the)g(e)o(xpansion)i Fn(I)6 b Fm(\()p Fn(x;)12 b(y)s(;)g(z)s Fm(\))21 b(=)h Fn(I)6 b Fm(\(\()p Fn(x;)11 b(y)s Fm(\))p Fn(;)i(z)s Fm(\))k(+)f Fn(I)6 b Fm(\()p Fn(x;)12 b(y)s Fm(\))p Fq(\).)-140 1053 y Fr(4.)45 b(KERNEL)18 b(INDEPENDENT)g (COMPONENT)g(AN)o(AL)-7 b(YSIS)-186 1202 y Fq(Ha)o(ving)28 b(de\002ned)g(a)g(contrast)g(function)g(in)g(terms)f(of)h(the)f (solution)i(of)e(a)-186 1289 y(generalized)i(eigen)m(v)n(alue)h (problem,)i(we)c(no)n(w)h(obtain)g(a)h(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)-186 1375 y(algorithm)21 b(by)h Fp(minimizing)f Fq(this)g(contrast)g(function)h(with)f(respect)g(to)g(the)-186 1462 y(parameter)e(matrix)g Fn(W)11 b Fq(.)-186 1657 y Fr(4.1.)45 b(Outline)17 b(of)i(algorithm)-186 1794 y Fq(Gi)n(v)o(en)26 b(a)f(set)h(of)f(data)h(v)o(ectors)g Fn(y)704 1763 y Fl(1)738 1794 y Fn(;)13 b(y)813 1763 y Fl(2)847 1794 y Fn(;)g(:)g(:)h(:)f(;)g(y)1059 1763 y Fk(N)1116 1794 y Fq(,)27 b(and)f(gi)n(v)o(en)g(a)g(parame-)-186 1881 y(ter)d(matrix)h Fn(W)11 b Fq(,)24 b(we)g(set)f Fn(x)523 1849 y Fk(i)580 1881 y Fm(=)30 b Fn(W)11 b(y)795 1849 y Fk(i)820 1881 y Fq(,)25 b(for)f(each)g Fn(i)p Fq(,)h(and)g(thereby)f(form)g(a)-186 1968 y(set)19 b(of)h(estimated)g (source)g(v)o(ectors)g Fi(f)p Fn(x)834 1936 y Fl(1)869 1968 y Fn(;)13 b(x)947 1936 y Fl(2)981 1968 y Fn(;)g(:)g(:)h(:)f(;)g(x) 1196 1936 y Fk(N)1253 1968 y Fi(g)p Fq(.)26 b(The)19 b Fn(m)g Fq(compo-)-186 2055 y(nents)27 b(of)g(these)h(v)o(ectors)f (yield)g(a)g(set)g(of)g Fn(m)f Fq(centered)i(Gram)f(matrices,)-186 2141 y Fn(K)-121 2149 y Fl(1)-86 2141 y Fn(;)13 b(K)13 2149 y Fl(2)47 2141 y Fn(;)h(:)f(:)g(:)g(;)g(K)283 2149 y Fk(m)342 2141 y Fq(.)50 b(These)29 b(Gram)f(matrices)g(\(which)g (depend)i(on)e Fn(W)11 b Fq(\))-186 2237 y(de\002ne)22 b(the)g(contrast)h(function,)g Fn(C)5 b Fm(\()p Fn(W)11 b Fm(\))26 b(=)1006 2218 y(^)997 2237 y Fn(I)1031 2245 y Fk(\032)1063 2256 y Ff(F)1116 2237 y Fm(\()p Fn(K)1211 2245 y Fl(1)1246 2237 y Fn(;)13 b(:)g(:)g(:)g(;)g(K)1481 2245 y Fk(m)1540 2237 y Fm(\))p Fq(,)22 b(as)g(the)-186 2323 y(solution)e(to)g(a)g(generalized)h(eigen)m(v)n(alue)h(problem,)e Fi(K)q Fn(\013)k Fm(=)f Fn(\025)p Fi(D)r Fn(\013)p Fq(,)d(where)-186 2410 y Fi(K)k Fq(and)g Fi(D)h Fq(are)e(block)h(matrices)f(constructed)i (from)e(the)g(Gram)g(matrices)-186 2497 y Fn(K)-121 2505 y Fk(i)-95 2497 y Fq(.)f(The)17 b(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)t(-)t(K)t(C)t(C)t(A)10 b(algorithm)16 b(in)m(v)o(olv)o(es) f(minimizing)g(this)-186 2584 y(function)20 b Fn(C)5 b Fm(\()p Fn(W)11 b Fm(\))17 b Fq(with)i(respect)g(to)g Fn(W)11 b Fq(.)-186 2764 y Fr(4.2.)45 b(Computational)18 b(issues)-186 2901 y Fq(In)i(order)h(to)g(turn)f(this)g(sk)o(etch)i (into)e(a)h(practical)f(ICA)g(algorithm,)h(se)n(v)o(eral)-186 2988 y(computational)f(issues)f(ha)o(v)o(e)h(to)e(be)i(addressed,)g(as) e(we)h(no)n(w)h(discuss.)-61 3075 y Fr(Numerical)h(linear)g(algebra.)31 b Fq(The)22 b Fi(F)7 b Fq(-correlation)22 b(in)m(v)o(olv)o(es)g(com-) -186 3161 y(puting)16 b(the)f(smallest)f(generalized)i(eigen)m(v)n (alue)h(of)e(matrices)g(of)g(size)g Fn(mN)8 b Fq(.)-186 3248 y(Thus)23 b(a)g(nai)n(v)o(e)h(implementation)g(w)o(ould)f(scale)g (as)g Fn(O)r Fm(\()p Fn(N)1340 3216 y Fl(3)1376 3248 y Fm(\))p Fq(,)g(a)g(computa-)-186 3335 y(tional)16 b(comple)o(xity)h (whose)f(cubic)h(gro)n(wth)g(in)e(the)i(number)g(of)f(data)g(points) -186 3422 y(w)o(ould)i(be)f(a)g(serious)g(liability)f(in)h (applications)h(to)f(lar)o(ge)g(data)g(sets.)22 b(Ho)n(w-)-186 3508 y(e)n(v)o(er)m(,)g(Gram)f(matrices)g(ha)o(v)o(e)g(a)g(spectrum)h (that)e(tends)i(to)f(sho)n(w)h(rapid)f(de-)-186 3595 y(cay)-5 b(,)20 b(and)g(lo)n(w-rank)h(approximations)g(of)f(Gram)f (matrices)h(can)g(therefore)-186 3682 y(often)26 b(pro)o(vide)i(suf)n (\002cient)e(\002delity)f(for)h(the)h(needs)g(of)f(k)o(ernel-based)i (al-)-186 3769 y(gorithms)h([10)q(].)52 b(In)29 b([3],)i(we)e(sho)n(w)g (theoretically)g(that)g(for)g(a)g(re)o(gular)o(-)-186 3856 y(ization)d(parameter)h Fn(\024)f Fq(that)g(is)f(linear)h(in)g Fn(N)8 b Fq(,)28 b(we)e(require)g(lo)n(w-rank)h(ap-)-186 3942 y(proximations)g(of)e(size)g Fn(M)8 b Fq(,)27 b(where)f Fn(M)33 b Fq(is)25 b(a)h(constant)g(that)f(is)g(indepen-)-186 4029 y(dent)g(of)f(the)h(number)h Fn(N)32 b Fq(of)25 b(samples.)40 b(Since)25 b(the)f(Gram)h(matrix)f Fn(K)1686 4037 y Fk(i)1737 4029 y Fq(is)-186 4116 y(positi)n(v)o(e)30 b(semide\002nite,)j(the)d(lo)n(w-rank)h(approximation)g(can)g(be)f (found)-186 4203 y(through)g(incomplete)f(Cholesk)o(y)h(decomposition)g (in)f(time)f Fn(O)r Fm(\()p Fn(M)1633 4171 y Fl(2)1668 4203 y Fn(N)8 b Fm(\))p Fq(,)-186 4289 y(which)28 b(gi)n(v)o(es)g(a)g Fn(M)k Fi(\002)24 b Fn(N)36 b Fq(matrix)27 b Fn(G)836 4297 y Fk(i)890 4289 y Fq(such)i(that)e Fn(K)1259 4297 y Fk(i)1324 4289 y Fi(\031)38 b Fn(G)1482 4297 y Fk(i)1508 4289 y Fn(G)1568 4258 y Fj(>)1568 4303 y Fk(i)1620 4289 y Fq(.)50 b(W)-6 b(e)-186 4376 y(perform)20 b(a)g(singular)g(v)n(alue)g (decomposition)h(of)f Fn(G)1167 4384 y Fk(i)1193 4376 y Fq(,)f(in)h(time)f Fn(O)r Fm(\()p Fn(M)1633 4344 y Fl(2)1668 4376 y Fn(N)8 b Fm(\))p Fq(,)-186 4463 y(to)19 b(obtain)g(an)g Fn(N)11 b Fi(\002)s Fn(M)27 b Fq(matrix)18 b Fn(U)686 4471 y Fk(i)731 4463 y Fq(with)g(orthogonal)j(columns)e (\(i.e.,)e(such)-186 4550 y(that)27 b Fn(U)13 4518 y Fj(>)5 4563 y Fk(i)66 4550 y Fn(U)118 4558 y Fk(i)182 4550 y Fm(=)37 b Fn(I)6 b Fq(\),)28 b(and)g(an)g Fn(M)19 b Fi(\002)11 b Fn(M)35 b Fq(diagonal)28 b(matrix)f Fm(\003)1456 4558 y Fk(i)1510 4550 y Fq(such)h(that)-186 4636 y Fn(K)-121 4644 y Fk(i)-73 4636 y Fi(\031)21 b Fn(G)68 4644 y Fk(i)94 4636 y Fn(G)154 4605 y Fj(>)154 4650 y Fk(i)228 4636 y Fm(=)g Fn(U)361 4644 y Fk(i)387 4636 y Fm(\003)440 4644 y Fk(i)467 4636 y Fn(U)527 4605 y Fj(>)519 4650 y Fk(i)580 4636 y Fq(.)-61 4723 y(W)-6 b(e)17 b(then)i(ha)o(v)o(e)g Fn(r)393 4731 y Fk(\024)433 4723 y Fm(\()p Fn(K)528 4731 y Fk(i)554 4723 y Fm(\))i(=)g(\()p Fn(K)781 4731 y Fk(i)822 4723 y Fm(+)14 b Fn(\024I)6 b Fm(\))1010 4691 y Fj(\000)p Fl(1)1093 4723 y Fn(K)1158 4731 y Fk(i)1205 4723 y Fm(=)21 b Fn(U)1338 4731 y Fk(i)1365 4723 y Fn(D)1428 4731 y Fk(i)1455 4723 y Fn(U)1515 4691 y Fj(>)1507 4737 y Fk(i)1567 4723 y Fq(,)d(where)-186 4810 y Fn(D)-123 4818 y Fk(i)-79 4810 y Fq(is)f(the)g(diagonal)i(matrix)e(obtained)h(from)g(the)f (diagonal)i(matrix)e Fm(\003)1668 4818 y Fk(i)1712 4810 y Fq(by)-186 4897 y(applying)28 b(the)e(function)h Fn(\025)36 b Fi(7!)f Fn(\025=)p Fm(\()p Fn(\025)22 b Fm(+)h Fn(\024)p Fm(\))j Fq(to)g(its)g(elements.)46 b(Finally)-5 b(,)-186 4983 y(in)26 b(the)h(tw)o(o-dimensional)g(case,)h(our)f(problem)g (reduces)g(to)f(\002nding)h(the)-186 5119 y(lar)o(gest)17 b(eigen)m(v)n(alue)h(of)461 5100 y Fe(e)443 5119 y Fi(R)508 5127 y Fk(\024)557 5119 y Fm(=)626 5014 y Fe(\022)876 5074 y Fm(0)224 b Fn(D)1201 5082 y Fl(1)1235 5074 y Fn(U)1295 5042 y Fj(>)1287 5087 y Fl(1)1348 5074 y Fn(U)1400 5082 y Fl(2)1435 5074 y Fn(D)1498 5082 y Fl(2)711 5167 y Fn(D)774 5175 y Fl(2)809 5167 y Fn(U)869 5135 y Fj(>)861 5180 y Fl(2)921 5167 y Fn(U)973 5175 y Fl(1)1008 5167 y Fn(D)1071 5175 y Fl(1)1329 5167 y Fm(0)1561 5014 y Fe(\023)1618 5119 y Fq(,)17 b(with)-186 5252 y(the)26 b(ob)o(vious)h(e)o(xtension)g (to)f(the)g Fn(m)p Fq(-dimensional)g(case.)45 b(This)25 b(problem)-186 5339 y(can)19 b(be)h(solv)o(ed)f(in)g(time)g(linear)f (in)h Fn(N)8 b Fq(.)2100 83 y Fr(Gradient)22 b(descent)g(on)h(the)f (Stiefel)g(manif)n(old.)34 b Fq(Since)23 b(decorrela-)1975 170 y(tion)g(implies)f(independence,)j(it)d(is)g(common)h(to)g(enforce) g(decorrelation)1975 257 y(of)h(the)f(estimated)h(sources.)37 b(This)23 b(is)g(done)i(by)f Fp(whitening)g Fq(the)f(data)h(and)1975 343 y(subsequently)k(restricting)e(the)g(minimization)g(to)g (orthogonal)i(matrices)1975 430 y Fn(W)37 b Fq([8)q(].)47 b(The)27 b(set)g(of)g(orthogonal)i(matrices,)f(which)g(is)e(commonly)j (re-)1975 517 y(ferred)d(to)f(as)g(the)h(Stiefel)e(manifold,)j(can)f (be)f(equipped)i(with)e(a)g(natural)1975 604 y(Riemannian)20 b(metric,)e(which)h(implies)g(that)g(gradient)g(algorithms)g(can)g(be) 1975 690 y(used.)24 b(In)19 b(our)g(simulations)g(we)g(used)g(steepest) g(descent)h(with)e(line)h(search)1975 777 y(along)27 b(geodesics.)46 b(The)26 b(algorithm)g(necessarily)h(con)m(v)o(er)o (ges)g(to)f(a)g(local)1975 864 y(minimum)20 b(of)f Fn(C)5 b Fm(\()p Fn(W)11 b Fm(\))p Fq(,)17 b(from)i(an)o(y)g(starting)g (point.)2100 953 y(The)h(ICA)f(contrast)h(functions)h(ha)o(v)o(e)f (multiple)g(local)f(minima,)h(ho)n(w-)1975 1040 y(e)n(v)o(er)m(,)c(and) f(restarts)f(are)g(generally)i(necessary)f(if)f(we)g(are)h(to)f(\002nd) h(the)f(global)1975 1127 y(optimum.)52 b(Empirically)-5 b(,)31 b(the)d(number)h(of)g(restarts)f(that)g(were)g(needed)1975 1213 y(w)o(as)21 b(found)h(to)e(be)h(small)f(when)i(the)e(number)i(of)e (samples)h(is)f(suf)n(\002ciently)1975 1300 y(lar)o(ge)25 b(so)h(as)f(to)g(mak)o(e)h(the)g(problem)g(well-de\002ned.)42 b(W)-6 b(e)25 b(ha)o(v)o(e)g(also)g(de-)1975 1387 y(v)o(eloped)32 b(tw)o(o)e(initialization)f(heuristics)h(that)g(ha)o(v)o(e)g(been)h (found)g(to)e(be)1975 1474 y(particularly)f(useful)g(in)f(practice)g (for)h(lar)o(ge-scale)f(problems,\223one-unit)1975 1560 y(contrast)h(functions\224,)h(and)f(Hermite)e(polynomial)j(k)o(ernels.) 48 b(These)27 b(are)1975 1647 y(detailed)20 b(in)e([3)q(].)1975 1841 y Fr(4.3.)45 b(K)n(er)o(nel)19 b(generalized)g(v)o(ariance)1975 1983 y Fq(The)k Fi(F)7 b Fq(-correlation)23 b(is)f(de\002ned)h(as)g (the)f(\002rst)g(eigen)m(v)n(alue)i(of)e(the)h(k)o(ernel-)1975 2070 y(ized)c(CCA)e(problem.)23 b(It)18 b(is)f(ob)o(viously)i(of)f (interest)g(to)g(consider)g(the)g(other)1975 2156 y(eigen)m(v)n(alues) 29 b(as)d(well.)46 b(Indeed,)29 b(there)e(is)f(a)h(classical)f (relationship)h(be-)1975 2243 y(tween)18 b(the)f(full)g(CCA)g(spectrum) h(and)g(the)f(mutual)h(information)g(of)f(Gaus-)1975 2330 y(sian)22 b(v)n(ariables)g Fn(x)2450 2338 y Fl(1)2506 2330 y Fq(and)g Fn(x)2679 2338 y Fl(2)2735 2330 y Fq([2]:)28 b(the)22 b(mutual)g(information)g Fn(I)6 b Fm(\()p Fn(x)3701 2338 y Fl(1)3735 2330 y Fn(;)13 b(x)3813 2338 y Fl(2)3847 2330 y Fm(\))21 b Fq(is)1975 2417 y(equal)h(to)f Fi(\000)2307 2386 y Fl(1)p 2306 2400 31 4 v 2306 2442 a(2)2360 2417 y Fm(log)2472 2361 y Fe(Q)2544 2439 y Fk(i)2570 2417 y Fm(\(1)e Fi(\000)g Fn(\032)2776 2385 y Fl(2)2776 2430 y Fk(i)2810 2417 y Fm(\))p Fq(.)28 b(The)21 b(product)3273 2361 y Fe(Q)3345 2439 y Fk(i)3372 2417 y Fm(\(1)e Fi(\000)f Fn(\032)3577 2385 y Fl(2)3577 2430 y Fk(i)3611 2417 y Fm(\))j Fq(is)f(usually)1975 2503 y(referred)g(to)e(as)h(the)g Fp(g)o(ener)o(alized)i(variance)p Fq(.)2100 2593 y(This)16 b(suggests)i(de\002ning)f(a)f(corresponding)j(quantity)e(for)f(k)o (ernelized)1975 2679 y(CCA.)23 b(In)h(the)g(case)g(of)g(tw)o(o)f(v)n (ariables,)j(we)d(de\002ne)h(the)g Fp(k)o(ernel)h(g)o(ener)o(al-)1975 2775 y(ized)k(variance)h(\(KGV\))e Fq(as)h(the)f(product)3113 2755 y Fm(^)3109 2775 y Fn(\016)3143 2783 y Fj(F)3236 2775 y Fm(=)3336 2719 y Fe(Q)3408 2797 y Fk(i)3435 2775 y Fm(\(1)c Fi(\000)h Fn(\032)3652 2743 y Fl(2)3652 2789 y Fk(i)3686 2775 y Fm(\))p Fq(,)30 b(where)1975 2861 y Fn(\032)2015 2869 y Fk(i)2065 2861 y Fq(are)23 b(the)h(\(positi)n(v)o (e\))f(k)o(ernel)i(canonical)f(correlations.)38 b(In)23 b(the)h(general)1975 2958 y(case)i(of)e Fn(m)h Fq(v)n(ariables,)h(we)f (de\002ne)2943 2938 y Fm(^)2939 2958 y Fn(\016)2973 2966 y Fj(F)3059 2958 y Fm(=)32 b(det)13 b Fi(K)q Fn(=)g Fm(det)g Fi(D)r Fq(.)41 b(Finally)-5 b(,)25 b(by)1975 3044 y(analogy)31 b(with)e(the)g(mutual)h(information)g(for)f(the)h(Gaussian)g(case,)i (we)1975 3140 y(also)e(de\002ne)g(a)g(contrast)g(function)2953 3121 y Fm(^)2944 3140 y Fn(I)2978 3149 y Fk(\016)3006 3160 y Ff(F)3101 3140 y Fm(=)42 b Fi(\000)3273 3110 y Fl(1)p 3272 3124 V 3272 3166 a(2)3325 3140 y Fm(log)3441 3121 y(^)3437 3140 y Fn(\016)3471 3148 y Fj(F)3525 3140 y Fq(.)55 b(It)29 b(turns)g(out)1975 3245 y(that)2123 3226 y Fm(^)2114 3245 y Fn(I)2148 3254 y Fk(\016)2176 3265 y Ff(F)2229 3245 y Fm(\()p Fn(K)2324 3253 y Fl(1)2358 3245 y Fn(;)13 b(:)h(:)f(:)g(;)g(K)2594 3253 y Fk(m)2653 3245 y Fm(\))26 b Fq(has)g(as)g(its)g(population)i(counterpart)f(a)f (func-)1975 3332 y(tion)f Fn(I)2150 3341 y Fk(\016)2178 3352 y Ff(F)2231 3332 y Fm(\()p Fn(x)2305 3340 y Fl(1)2339 3332 y Fn(;)13 b(:)g(:)h(:)f(;)g(x)2554 3340 y Fk(m)2612 3332 y Fm(\))24 b Fq(that)g(is)g(an)h(approximation)h(of)f(the)f (mutual)h(in-)1975 3418 y(formation)17 b(between)f(the)g(original)f (non-Gaussian)j(v)n(ariables)e(in)g(the)f(input)1975 3505 y(space)20 b([3].)2494 3707 y Fr(5.)45 b(SIMULA)-7 b(TION)17 b(RESUL)-7 b(TS)1975 3861 y Fq(W)h(e)32 b(ha)o(v)o(e)h (conducted)h(an)e(e)o(xtensi)n(v)o(e)h(set)f(of)g(simulation)h(e)o (xperiments)1975 3948 y(using)d(data)f(obtained)h(from)e(a)h(v)n (ariety)g(of)g(source)g(distrib)o(utions.)53 b(The)1975 4035 y(sources)19 b(that)f(we)h(used)f(\(Figure)g(1,)h(T)-6 b(op\))18 b(included)h(subgaussian)h(and)f(su-)1975 4121 y(per)o(gaussian)28 b(distrib)o(utions,)h(as)d(well)h(as)g(distrib)o (utions)g(that)f(are)h(nearly)1975 4208 y(Gaussian.)c(W)-6 b(e)17 b(studied)g(unimodal,)h(multimodal,)f(symmetric,)g(and)h(non-) 1975 4295 y(symmetric)29 b(distrib)o(utions.)53 b(W)-6 b(e)29 b(also)g(v)n(aried)g(the)g(number)h(of)f(compo-)1975 4382 y(nents,)24 b(from)f(2)g(to)f(16,)i(the)f(number)h(of)f(training)g (samples,)g(from)g(250)h(to)1975 4469 y(4000,)18 b(and)f(studied)g(the) f(rob)o(ustness)h(of)f(the)g(algorithms)h(to)f(v)n(arying)h(num-)1975 4555 y(bers)j(of)f(outliers)f(\(see)h([3)q(])f(for)h(details\).)2100 4645 y(Comparisons)24 b(were)f(made)h(with)e(three)i(e)o(xisting)f(ICA) f(algorithms:)1975 4731 y(the)15 b(F)o(astICA)e(algorithm)i([8],)g(the) g(Jade)g(algorithm)g([7],)g(and)g(the)f(e)o(xtended)1975 4818 y(Infomax)21 b(algorithm)f([9].)26 b(All)19 b(simulations)h(were)f (performed)i(in)f(the)f(sit-)1975 4905 y(uation)j(when)f(the)g(true)g (demixing)h(matrix)f Fn(W)3206 4913 y Fl(0)3261 4905 y Fq(is)f(kno)n(wn.)30 b(W)-6 b(e)21 b(measure)1975 4992 y(the)29 b(performance)h(of)f(the)g(algorithm)g(in)g(terms)f(of)h(the)g (dif)n(ference)g(be-)1975 5078 y(tween)21 b Fn(W)30 b Fq(and)21 b Fn(W)2478 5086 y Fl(0)2512 5078 y Fq(,)g(via)f(the)g (standard)i(ICA)d(metric)i(introduced)g(by)g([1].)1975 5165 y(This)k(measure)i(is)e(in)m(v)n(ariant)h(to)f(permutation)h(and)g (scaling)g(of)g(its)f(ar)o(gu-)1975 5252 y(ments,)i(lies)e(between)h Fm(0)g Fq(and)g Fm(100\()p Fn(m)d Fi(\000)e Fm(1\))p Fq(,)27 b(and)f(is)f(equal)h(to)f(zero)h(for)1975 5339 y(perfect)20 b(demixing.)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 63 628 a @beginspecial 174 @llx 338 @lly 451 @urx 456 @ury 1770 @rwi @setspecial %%BeginDocument: ICAdistributions.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: C:\KernelICA\Tex - ICASSP\ICAdistributions.eps %%CreationDate: 10/10/2002 18:13:47 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 174 338 451 456 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /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 /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /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 % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} 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 {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 174 338 451 456 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 1584 5532 csm 508 50 3321 1422 MR c np 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 4177 1585 PR 3 w 0 354 425 0 0 -354 543 473 4 MP PP -425 0 0 354 425 0 0 -354 543 473 5 MP stroke 2 w DO SO 3 w 0 sg 543 119 mt 968 119 L 543 473 mt 968 473 L 968 473 mt 968 119 L 543 473 mt 543 119 L 543 473 mt 968 473 L 543 473 mt 543 119 L 613 473 mt 613 468 L 613 119 mt 613 123 L 755 473 mt 755 468 L 755 119 mt 755 123 L 897 473 mt 897 468 L 897 119 mt 897 123 L 543 473 mt 547 473 L 968 473 mt 963 473 L 543 220 mt 547 220 L 968 220 mt 963 220 L 543 119 mt 968 119 L 543 473 mt 968 473 L 968 473 mt 968 119 L 543 473 mt 543 119 L gs 543 119 426 355 MR c np 6 w 2 0 1 1 2 0 1 0 2 0 1 1 1 0 2 0 1 0 2 1 1 0 1 0 2 1 1 0 2 0 1 1 2 0 1 0 1 1 2 0 1 0 2 1 1 0 1 1 2 0 1 0 2 1 1 0 2 1 1 0 1 1 2 0 1 1 2 0 1 1 1 0 2 1 1 1 2 0 1 1 2 0 1 1 1 1 2 0 1 1 2 1 1 0 1 1 2 1 1 1 2 0 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 0 1 1 2 1 1 1 2 2 1 1 2 1 1 1 1 1 2 1 1 1 2 2 1 1 1 1 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 1 1 1 2 2 2 1 1 2 2 1 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 1 2 2 2 1 2 2 2 827 370 100 MP stroke 1 2 2 2 1 2 1 2 2 3 1 2 2 2 1 2 1 2 2 3 1 2 2 2 1 2 2 2 1 3 1 2 2 2 1 2 2 2 1 3 1 2 2 2 1 2 2 2 1 2 2 2 1 2 1 2 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 1 1 1 1 2 2 1 1 0 2 1 1 1 1 1 2 0 1 1 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 -1 1 0 2 -1 1 -1 2 -1 1 0 2 -1 1 -2 1 -1 2 -1 1 -1 2 -2 1 -1 1 -2 2 -1 1 -2 2 -2 1 -1 2 -2 1 -2 1 -2 2 -2 1 -2 2 -2 1 -2 1 -2 2 -2 1 -3 2 -2 1 -2 2 -2 1 -2 1 -3 2 -2 1 -2 2 -2 1 -2 1 -3 2 -2 1 -2 2 -2 1 -2 2 -3 1 -2 687 364 100 MP stroke 1 -2 2 -2 1 -2 2 -2 1 -2 1 -2 2 -2 1 -2 2 -2 1 -2 2 -2 1 -2 1 -1 2 -2 1 -2 2 -2 1 -1 1 -2 2 -2 1 -1 2 -2 1 -2 2 -1 1 -2 1 -1 2 -1 1 -2 2 -1 1 -2 1 -1 2 -1 1 -1 2 -2 1 -1 2 -1 1 -1 1 -1 2 -1 1 -1 2 -2 1 -1 1 -1 2 -1 1 0 2 -1 1 -1 2 -1 1 -1 1 -1 2 -1 1 -1 2 0 1 -1 1 -1 2 -1 1 0 2 -1 1 -1 2 0 1 -1 1 -1 2 0 1 -1 2 0 1 -1 1 -1 2 0 1 -1 2 0 1 -1 2 0 1 -1 1 0 2 -1 1 0 2 -1 1 0 1 0 2 -1 1 0 2 -1 1 0 2 0 1 -1 1 0 2 0 1 -1 2 0 1 0 1 -1 2 0 1 0 2 -1 1 0 2 0 1 0 1 -1 2 0 1 0 547 460 100 MP stroke 2 0 1 -1 1 0 543 461 4 MP stroke gr 6 w %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 562 211 mt (\(a\)) s 3 w 1 sg 0 354 426 0 0 -354 1105 473 4 MP PP -426 0 0 354 426 0 0 -354 1105 473 5 MP stroke 2 w DO SO 3 w 0 sg 1105 119 mt 1531 119 L 1105 473 mt 1531 473 L 1531 473 mt 1531 119 L 1105 473 mt 1105 119 L 1105 473 mt 1531 473 L 1105 473 mt 1105 119 L 1176 473 mt 1176 468 L 1176 119 mt 1176 123 L 1318 473 mt 1318 468 L 1318 119 mt 1318 123 L 1460 473 mt 1460 468 L 1460 119 mt 1460 123 L 1105 473 mt 1109 473 L 1531 473 mt 1526 473 L 1105 312 mt 1109 312 L 1531 312 mt 1526 312 L 1105 151 mt 1109 151 L 1531 151 mt 1526 151 L 1105 119 mt 1531 119 L 1105 473 mt 1531 473 L 1531 473 mt 1531 119 L 1105 473 mt 1105 119 L gs 1105 119 427 355 MR c np 6 w 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 1 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 1 2 0 1 0 1 0 2 0 1 0 2 1 1 0 2 0 1 0 1 0 2 1 1 0 2 0 1 0 2 0 1 1 1 0 2 0 1 0 2 1 1 0 1 0 2 0 1 1 2 0 1 0 2 1 1 0 1 0 2 1 1 0 2 0 1 1 1 0 2 0 1 1 2 0 1 1 2 0 1 0 1 1 2 0 1 1 2 0 1 1 2 0 1 1 1 0 2 1 1 1 2 0 1 1 1 1 2 0 1 1 2 1 1 0 2 1 1 1 1 1 2 0 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 1 1 2 2 1 1 1390 419 100 MP stroke 1 2 2 1 1 2 2 2 1 1 2 2 1 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 1 3 2 2 1 2 2 3 1 3 1 2 2 3 1 3 2 3 1 3 2 3 1 3 1 3 2 3 1 4 2 3 1 4 2 3 1 4 1 4 2 4 1 4 2 5 1 4 1 5 2 4 1 5 2 5 1 5 2 5 1 6 1 5 2 6 1 6 2 6 1 6 1 6 2 -6 1 -6 2 -6 1 -6 2 -6 1 -5 1 -6 2 -5 1 -5 2 -5 1 -5 2 -4 1 -5 1 -4 2 -5 1 -4 2 -4 1 -4 1 -4 2 -3 1 -4 2 -3 1 -4 2 -3 1 -3 1 -3 2 -3 1 -3 2 -3 1 -3 2 -3 1 -2 1 -3 2 -3 1 -2 2 -2 1 -3 1 -2 2 -2 1 -2 2 -2 1 -2 2 -2 1 -2 1 -2 2 -2 1 -1 2 -2 1249 414 100 MP stroke 1 -2 1 -1 2 -2 1 -1 2 -2 1 -1 2 -1 1 -2 1 -1 2 -1 1 -2 2 -1 1 -1 2 -1 1 -1 1 -1 2 -1 1 -1 2 -1 1 -1 1 -1 2 -1 1 -1 2 -1 1 -1 2 0 1 -1 1 -1 2 -1 1 0 2 -1 1 -1 2 0 1 -1 1 -1 2 0 1 -1 2 -1 1 0 1 -1 2 0 1 -1 2 0 1 -1 2 0 1 -1 1 0 2 0 1 -1 2 0 1 -1 1 0 2 0 1 -1 2 0 1 0 2 -1 1 0 1 0 2 -1 1 0 2 0 1 -1 2 0 1 0 1 0 2 -1 1 0 2 0 1 0 1 -1 2 0 1 0 2 0 1 0 2 -1 1 0 1 0 2 0 1 0 2 -1 1 0 2 0 1 0 1 0 2 0 1 -1 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 -1 1 0 1 0 2 0 1 0 1109 469 100 MP stroke 2 0 1 0 1 0 1105 469 4 MP stroke gr 6 w 1124 209 mt (\(b\)) s 3 w 1 sg 0 354 426 0 0 -354 1667 473 4 MP PP -426 0 0 354 426 0 0 -354 1667 473 5 MP stroke 2 w DO SO 3 w 0 sg 1667 119 mt 2093 119 L 1667 473 mt 2093 473 L 2093 473 mt 2093 119 L 1667 473 mt 1667 119 L 1667 473 mt 2093 473 L 1667 473 mt 1667 119 L 1738 473 mt 1738 468 L 1738 119 mt 1738 123 L 1880 473 mt 1880 468 L 1880 119 mt 1880 123 L 2022 473 mt 2022 468 L 2022 119 mt 2022 123 L 1667 473 mt 1671 473 L 2093 473 mt 2088 473 L 1667 220 mt 1671 220 L 2093 220 mt 2088 220 L 1667 119 mt 2093 119 L 1667 473 mt 2093 473 L 2093 473 mt 2093 119 L 1667 473 mt 1667 119 L gs 1667 119 427 355 MR c np 6 w 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 146 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1952 327 100 MP stroke 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1811 327 100 MP stroke 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 -146 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 1671 473 100 MP stroke 2 0 1 0 1 0 1667 473 4 MP stroke gr 6 w 1686 211 mt (\(c\)) s 3 w 1 sg 0 354 426 0 0 -354 2229 473 4 MP PP -426 0 0 354 426 0 0 -354 2229 473 5 MP stroke 2 w DO SO 3 w 0 sg 2229 119 mt 2655 119 L 2229 473 mt 2655 473 L 2655 473 mt 2655 119 L 2229 473 mt 2229 119 L 2229 473 mt 2655 473 L 2229 473 mt 2229 119 L 2300 473 mt 2300 468 L 2300 119 mt 2300 123 L 2442 473 mt 2442 468 L 2442 119 mt 2442 123 L 2584 473 mt 2584 468 L 2584 119 mt 2584 123 L 2229 473 mt 2233 473 L 2655 473 mt 2650 473 L 2229 220 mt 2233 220 L 2655 220 mt 2650 220 L 2229 119 mt 2655 119 L 2229 473 mt 2655 473 L 2655 473 mt 2655 119 L 2229 473 mt 2229 119 L gs 2229 119 427 355 MR c np 6 w 2 0 1 1 2 0 1 0 2 0 1 1 1 0 2 0 1 1 2 0 1 0 2 0 1 1 1 0 2 0 1 1 2 0 1 0 1 1 2 0 1 1 2 0 1 0 2 1 1 0 1 1 2 0 1 1 2 0 1 1 2 0 1 1 1 0 2 1 1 1 2 0 1 1 1 0 2 1 1 1 2 0 1 1 2 1 1 1 1 0 2 1 1 1 2 1 1 1 1 0 2 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 2 2 1 1 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 1 1 2 2 1 1 2 2 1 1 1 2 2 2 1 1 2 2 1 2 2 1 1 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 2 1 3 1 2 2 2 1 2 2 2 1 2 2514 364 100 MP stroke 1 3 2 2 1 2 2 2 1 3 2 2 1 2 1 2 2 3 1 2 2 2 1 3 2 2 1 2 1 2 2 3 1 2 2 2 1 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 1 2 2 2 1 2 2 2 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 1 2 1 1 1 2 1 1 1 2 0 1 1 1 0 2 1 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 2 -1 1 0 1 -1 2 0 1 -1 2 -1 1 -1 2 -1 1 -1 1 -1 2 -2 1 -1 2 -1 1 -2 1 -1 2 -2 1 -1 2 -2 1 -2 2 -2 1 -2 1 -2 2 -2 1 -2 2 -2 1 -2 2 -2 1 -2 1 -2 2 -2 1 -2 2 -3 1 -2 1 -2 2 -2 1 -3 2 -2 1 -2 2 -3 1 -2 1 -2 2 -2 1 -3 2 -2 2373 357 100 MP stroke 1 -2 1 -2 2 -3 1 -2 2 -2 1 -2 2 -2 1 -2 1 -3 2 -2 1 -2 2 -2 1 -2 2 -2 1 -2 1 -1 2 -2 1 -2 2 -2 1 -2 1 -2 2 -1 1 -2 2 -2 1 -1 2 -2 1 -2 1 -1 2 -2 1 -1 2 -2 1 -1 2 -1 1 -2 1 -1 2 -1 1 -2 2 -1 1 -1 1 -1 2 -1 1 -2 2 -1 1 -1 2 -1 1 -1 1 -1 2 -1 1 -1 2 -1 1 -1 1 -1 2 0 1 -1 2 -1 1 -1 2 -1 1 0 1 -1 2 -1 1 -1 2 0 1 -1 2 -1 1 0 1 -1 2 0 1 -1 2 -1 1 0 1 -1 2 0 1 -1 2 0 1 -1 2 0 1 -1 1 0 2 -1 1 0 2 0 1 -1 2 0 1 -1 1 0 2 0 1 -1 2 0 1 0 1 -1 2 0 1 0 2 0 1 -1 2 0 1 0 1 -1 2 0 1 0 2233 463 100 MP stroke 2 0 1 -1 1 0 2229 464 4 MP stroke gr 6 w 2248 211 mt (\(d\)) s 3 w 1 sg 0 354 425 0 0 -354 2792 473 4 MP PP -425 0 0 354 425 0 0 -354 2792 473 5 MP stroke 2 w DO SO 3 w 0 sg 2792 119 mt 3217 119 L 2792 473 mt 3217 473 L 3217 473 mt 3217 119 L 2792 473 mt 2792 119 L 2792 473 mt 3217 473 L 2792 473 mt 2792 119 L 2792 473 mt 2792 468 L 2792 119 mt 2792 123 L 3004 473 mt 3004 468 L 3004 119 mt 3004 123 L 3217 473 mt 3217 468 L 3217 119 mt 3217 123 L 2792 473 mt 2796 473 L 3217 473 mt 3212 473 L 2792 346 mt 2796 346 L 3217 346 mt 3212 346 L 2792 220 mt 2796 220 L 3217 220 mt 3212 220 L 2792 119 mt 3217 119 L 2792 473 mt 3217 473 L 3217 473 mt 3217 119 L 2792 473 mt 2792 119 L gs 2792 119 426 355 MR c np 6 w 1 0 3 0 2 1 2 0 2 0 2 0 2 1 2 0 2 0 3 1 2 0 2 0 2 1 2 0 2 0 2 0 2 1 3 0 2 1 2 0 2 0 2 1 2 0 2 0 2 1 3 0 2 1 2 0 2 1 2 0 2 0 2 1 2 0 3 1 2 0 2 1 2 0 2 1 2 0 2 1 2 1 3 0 2 1 2 0 2 1 2 0 2 1 2 1 2 0 3 1 2 1 2 0 2 1 2 1 2 1 2 0 2 1 3 1 2 1 2 0 2 1 2 1 2 1 2 1 2 1 3 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 3 1 2 1 2 1 2 1 2 1 2 1 2 2 2 1 3 1 2 1 2 2 2 1 2 1 2 2 2 1 2 2 3 1 2 2 2 1 2 2 2 1 2 2 2 2 2 1 3 2 2 2 3008 383 100 MP stroke 2 2 2 2 2 1 2 2 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 3 2 2 3 2 2 2 3 2 2 2 3 2 3 2 2 3 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 3 4 2 3 2 3 2 4 2 3 2 4 2 4 2 4 3 3 2 4 2 4 2 5 2 4 2 4 2 4 2 5 3 4 2 5 2 5 2 4 2 5 2 -248 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2798 473 100 MP stroke 2 0 2 0 2 0 2792 473 4 MP stroke gr 6 w 2813 211 mt (\(e\)) s 3 w 1 sg 0 354 425 0 0 -354 3354 473 4 MP PP -425 0 0 354 425 0 0 -354 3354 473 5 MP stroke 2 w DO SO 3 w 0 sg 3354 119 mt 3779 119 L 3354 473 mt 3779 473 L 3779 473 mt 3779 119 L 3354 473 mt 3354 119 L 3354 473 mt 3779 473 L 3354 473 mt 3354 119 L 3424 473 mt 3424 468 L 3424 119 mt 3424 123 L 3566 473 mt 3566 468 L 3566 119 mt 3566 123 L 3708 473 mt 3708 468 L 3708 119 mt 3708 123 L 3354 473 mt 3358 473 L 3779 473 mt 3774 473 L 3354 296 mt 3358 296 L 3779 296 mt 3774 296 L 3354 119 mt 3358 119 L 3779 119 mt 3774 119 L 3354 119 mt 3779 119 L 3354 473 mt 3779 473 L 3779 473 mt 3779 119 L 3354 473 mt 3354 119 L gs 3354 119 426 355 MR c np 6 w 2 0 1 0 2 1 1 0 2 0 1 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 2 1 1 0 2 0 1 0 1 0 2 0 1 0 2 1 1 0 1 0 2 0 1 0 2 1 1 0 2 0 1 0 1 1 2 0 1 0 2 1 1 0 1 0 2 1 1 0 2 0 1 1 2 0 1 1 1 0 2 1 1 1 2 0 1 1 1 1 2 1 1 0 2 1 1 1 2 1 1 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 1 1 2 2 2 1 2 1 2 2 3 1 2 2 3 1 2 1 3 2 3 1 3 2 4 1 3 2 4 1 4 1 4 2 5 1 4 2 5 1 6 1 5 2 6 1 6 2 7 1 7 2 7 1 8 1 8 2 9 1 10 2 9 1 11 1 11 2 11 1 13 2 13 3638 235 100 MP stroke 1 14 2 -14 1 -13 1 -12 2 -12 1 -11 2 -10 1 -10 1 -9 2 -9 1 -8 2 -8 1 -7 2 -7 1 -6 1 -6 2 -6 1 -6 2 -5 1 -4 1 -5 2 -4 1 -4 2 -4 1 -4 2 -3 1 -3 1 -3 2 -3 1 -2 2 -3 1 -2 1 -2 2 -2 1 -2 2 -2 1 -1 2 -2 1 -1 1 -1 2 -1 1 -1 2 -1 1 -1 1 0 2 -1 1 0 2 0 1 -1 2 0 1 0 1 0 2 0 1 1 2 0 1 0 1 1 2 0 1 1 2 1 1 1 2 1 1 1 1 1 2 2 1 1 2 2 1 2 1 2 2 2 1 2 2 3 1 2 2 3 1 3 1 3 2 3 1 4 2 4 1 4 1 4 2 5 1 4 2 5 1 6 2 6 1 6 1 6 2 7 1 7 2 8 1 8 1 9 2 9 1 10 2 10 1 11 2 12 1 12 3498 248 100 MP stroke 1 13 2 14 1 -14 2 -13 1 -13 1 -11 2 -11 1 -11 2 -9 1 -10 2 -9 1 -8 1 -8 2 -7 1 -7 2 -7 1 -6 1 -6 2 -5 1 -6 2 -5 1 -4 2 -5 1 -4 1 -4 2 -4 1 -3 2 -4 1 -3 1 -3 2 -3 1 -2 2 -3 1 -2 2 -3 1 -2 1 -2 2 -2 1 -2 2 -1 1 -2 1 -1 2 -2 1 -1 2 -1 1 -2 2 -1 1 -1 1 -1 2 -1 1 -1 2 -1 1 0 1 -1 2 -1 1 -1 2 0 1 -1 2 -1 1 0 1 -1 2 0 1 -1 2 0 1 0 1 -1 2 0 1 0 2 -1 1 0 2 0 1 -1 1 0 2 0 1 0 2 -1 1 0 1 0 2 0 1 0 2 -1 1 0 2 0 1 0 1 0 2 0 1 0 2 -1 1 0 1 0 2 0 1 0 2 0 1 0 2 0 1 0 1 0 2 0 1 0 3358 471 100 MP stroke 2 -1 1 0 1 0 3354 472 4 MP stroke gr 6 w 3373 217 mt (\(f\)) s 3 w 1 sg 0 355 425 0 0 -355 543 942 4 MP PP -425 0 0 355 425 0 0 -355 543 942 5 MP stroke 2 w DO SO 3 w 0 sg 543 587 mt 968 587 L 543 942 mt 968 942 L 968 942 mt 968 587 L 543 942 mt 543 587 L 543 942 mt 968 942 L 543 942 mt 543 587 L 543 942 mt 543 937 L 543 587 mt 543 591 L 755 942 mt 755 937 L 755 587 mt 755 591 L 968 942 mt 968 937 L 968 587 mt 968 591 L 543 942 mt 547 942 L 968 942 mt 963 942 L 543 755 mt 547 755 L 968 755 mt 963 755 L 543 587 mt 968 587 L 543 942 mt 968 942 L 968 942 mt 968 587 L 543 942 mt 543 587 L gs 543 587 426 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 1 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 1 2 1 2 1 2 1 2 3 2 3 3 3 2 6 2 6 2 8 2 10 2 12 2 14 2 15 3 18 2 18 2 20 2 20 2 20 2 18 2 17 2 14 3 10 2 7 2 2 2 -2 2 -7 2 -10 2 -14 2 -17 3 -18 2 -20 2 -20 2 -20 2 -18 2 -18 2 -15 2 -14 3 -12 2 -10 2 -8 2 -6 2 -6 2 -3 2 -3 2 -2 3 -2 2 -1 759 939 100 MP stroke 2 0 2 -1 2 1 2 0 2 1 2 2 3 2 2 3 2 3 2 6 2 6 2 8 2 10 2 12 3 14 2 15 2 18 2 18 2 20 2 20 2 20 2 18 3 17 2 14 2 10 2 7 2 2 2 -2 2 -7 2 -10 3 -14 2 -17 2 -18 2 -20 2 -20 2 -20 2 -18 2 -18 3 -15 2 -14 2 -12 2 -10 2 -8 2 -6 2 -6 2 -3 3 -3 2 -3 2 -1 2 -1 2 -1 2 -1 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 -1 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 549 942 100 MP stroke 2 0 2 0 2 0 543 942 4 MP stroke gr 6 w 564 679 mt (\(g\)) s 3 w 1 sg 0 355 426 0 0 -355 1105 942 4 MP PP -426 0 0 355 426 0 0 -355 1105 942 5 MP stroke 2 w DO SO 3 w 0 sg 1105 587 mt 1531 587 L 1105 942 mt 1531 942 L 1531 942 mt 1531 587 L 1105 942 mt 1105 587 L 1105 942 mt 1531 942 L 1105 942 mt 1105 587 L 1105 942 mt 1105 937 L 1105 587 mt 1105 591 L 1318 942 mt 1318 937 L 1318 587 mt 1318 591 L 1531 942 mt 1531 937 L 1531 587 mt 1531 591 L 1105 942 mt 1109 942 L 1531 942 mt 1526 942 L 1105 744 mt 1109 744 L 1531 744 mt 1526 744 L 1105 587 mt 1531 587 L 1105 942 mt 1531 942 L 1531 942 mt 1531 587 L 1105 942 mt 1105 587 L gs 1105 587 427 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 1 2 0 2 0 2 0 2 0 2 1 3 0 2 0 2 1 2 0 2 1 2 1 2 0 2 1 3 1 2 1 2 1 2 2 2 1 2 1 2 2 3 2 2 2 2 2 2 3 2 2 2 3 2 3 2 3 3 3 2 4 2 4 2 4 2 4 2 5 2 4 2 5 3 5 2 6 2 5 2 6 2 5 2 6 2 6 3 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 3 5 2 5 2 5 2 5 2 5 2 4 2 4 2 3 3 4 2 2 2 3 2 2 2 2 2 1 2 1 3 0 2 0 2 0 2 0 2 -1 2 -1 2 -1 2 -2 3 -1 2 -2 2 -2 2 -2 2 -2 2 -1 2 -2 2 -2 3 -2 2 -1 2 -1 2 -1 2 -1 2 -1 1322 761 100 MP stroke 2 0 2 0 3 0 2 0 2 1 2 1 2 1 2 1 2 1 3 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 3 1 2 2 2 1 2 1 2 1 2 0 2 0 2 0 3 0 2 -1 2 -1 2 -2 2 -2 2 -3 2 -2 3 -4 2 -3 2 -4 2 -4 2 -5 2 -5 2 -5 2 -5 3 -5 2 -6 2 -6 2 -6 2 -6 2 -6 2 -6 2 -6 3 -6 2 -6 2 -6 2 -5 2 -6 2 -5 2 -6 3 -5 2 -5 2 -4 2 -5 2 -4 2 -4 2 -4 2 -4 3 -3 2 -3 2 -3 2 -3 2 -2 2 -3 2 -2 2 -2 3 -2 2 -2 2 -1 2 -1 2 -2 2 -1 2 -1 3 -1 2 -1 2 0 2 -1 2 -1 2 0 2 -1 2 0 3 0 2 -1 2 0 2 0 2 0 2 0 2 -1 2 0 3 0 2 0 2 0 2 0 2 0 1111 941 100 MP stroke 2 0 2 0 2 0 1105 941 4 MP stroke gr 6 w 1126 680 mt (\(h\)) s 3 w 1 sg 0 355 426 0 0 -355 1667 942 4 MP PP -426 0 0 355 426 0 0 -355 1667 942 5 MP stroke 2 w DO SO 3 w 0 sg 1667 587 mt 2093 587 L 1667 942 mt 2093 942 L 2093 942 mt 2093 587 L 1667 942 mt 1667 587 L 1667 942 mt 2093 942 L 1667 942 mt 1667 587 L 1667 942 mt 1667 937 L 1667 587 mt 1667 591 L 1880 942 mt 1880 937 L 1880 587 mt 1880 591 L 2093 942 mt 2093 937 L 2093 587 mt 2093 591 L 1667 942 mt 1671 942 L 2093 942 mt 2088 942 L 1667 744 mt 1671 744 L 2093 744 mt 2088 744 L 1667 587 mt 2093 587 L 1667 942 mt 2093 942 L 2093 942 mt 2093 587 L 1667 942 mt 1667 587 L gs 1667 587 427 356 MR c np 6 w 1 0 3 0 2 1 2 0 2 0 2 1 2 0 2 0 3 1 2 0 2 1 2 0 2 1 2 1 2 1 2 0 3 1 2 1 2 1 2 1 2 2 2 1 2 1 2 2 3 1 2 2 2 2 2 1 2 2 2 3 2 2 3 2 2 3 2 2 2 3 2 2 2 3 2 3 2 4 3 3 2 3 2 3 2 4 2 4 2 3 2 4 2 4 3 4 2 4 2 4 2 4 2 4 2 4 2 4 3 4 2 4 2 4 2 4 2 4 2 4 2 3 2 4 3 4 2 3 2 3 2 4 2 3 2 3 2 3 2 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2 1 2 1 2 1 2 1 2 0 2 1 3 1 2 0 2 0 2 1 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 1884 751 100 MP stroke 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 -1 2 0 2 0 3 -1 2 -1 2 0 2 -1 2 -1 2 -1 2 -1 2 -2 3 -1 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 3 -3 2 -2 2 -3 2 -3 2 -3 2 -4 2 -3 2 -3 3 -4 2 -4 2 -3 2 -4 2 -4 2 -4 2 -4 2 -4 3 -4 2 -4 2 -4 2 -4 2 -4 2 -4 2 -4 3 -4 2 -4 2 -4 2 -3 2 -4 2 -4 2 -3 2 -3 3 -3 2 -4 2 -3 2 -3 2 -2 2 -3 2 -2 2 -3 3 -2 2 -2 2 -3 2 -2 2 -1 2 -2 2 -2 3 -1 2 -2 2 -1 2 -1 2 -2 2 -1 2 -1 2 -1 3 -1 2 0 2 -1 2 -1 2 -1 2 0 2 -1 2 0 3 -1 2 0 2 0 2 -1 2 0 1673 939 100 MP stroke 2 0 2 -1 2 0 1667 940 4 MP stroke gr 6 w 1688 680 mt (\(i\)) s 3 w 1 sg 0 355 426 0 0 -355 2229 942 4 MP PP -426 0 0 355 426 0 0 -355 2229 942 5 MP stroke 2 w DO SO 3 w 0 sg 2229 587 mt 2655 587 L 2229 942 mt 2655 942 L 2655 942 mt 2655 587 L 2229 942 mt 2229 587 L 2229 942 mt 2655 942 L 2229 942 mt 2229 587 L 2229 942 mt 2229 937 L 2229 587 mt 2229 591 L 2442 942 mt 2442 937 L 2442 587 mt 2442 591 L 2655 942 mt 2655 937 L 2655 587 mt 2655 591 L 2229 942 mt 2233 942 L 2655 942 mt 2650 942 L 2229 810 mt 2233 810 L 2655 810 mt 2650 810 L 2229 679 mt 2233 679 L 2655 679 mt 2650 679 L 2229 587 mt 2655 587 L 2229 942 mt 2655 942 L 2655 942 mt 2655 587 L 2229 942 mt 2229 587 L gs 2229 587 427 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 1 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 1 2 0 2 1 3 1 2 2 2 2 2 4 2 4 2 5 2 6 2 8 3 9 2 11 2 13 2 13 2 15 2 16 2 16 2 15 3 15 2 13 2 11 2 8 2 5 2 2 2 -2 3 -5 2 -8 2 -11 2 -13 2 -15 2 -15 2 -16 2 -16 3 -15 2 -13 2 -13 2 -11 2 -9 2 -8 2 -6 2 -5 3 -4 2 -4 2 -2 2 -2 2 -1 2 -1 2446 940 100 MP stroke 2 0 2 -1 3 0 2 0 2 0 2 0 2 0 2 0 2 1 3 0 2 1 2 0 2 1 2 1 2 2 2 1 2 1 3 2 2 2 2 2 2 1 2 2 2 1 2 2 2 1 3 0 2 0 2 0 2 0 2 -1 2 -2 2 -1 3 -2 2 -1 2 -2 2 -2 2 -2 2 -1 2 -1 2 -2 3 -1 2 -1 2 0 2 -1 2 0 2 -1 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 -1 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2235 942 100 MP stroke 2 0 2 0 2 0 2229 942 4 MP stroke gr 6 w 2250 676 mt (\(j\)) s 3 w 1 sg 0 355 425 0 0 -355 2792 942 4 MP PP -425 0 0 355 425 0 0 -355 2792 942 5 MP stroke 2 w DO SO 3 w 0 sg 2792 587 mt 3217 587 L 2792 942 mt 3217 942 L 3217 942 mt 3217 587 L 2792 942 mt 2792 587 L 2792 942 mt 3217 942 L 2792 942 mt 2792 587 L 2792 942 mt 2792 937 L 2792 587 mt 2792 591 L 3004 942 mt 3004 937 L 3004 587 mt 3004 591 L 3217 942 mt 3217 937 L 3217 587 mt 3217 591 L 2792 942 mt 2796 942 L 3217 942 mt 3212 942 L 2792 744 mt 2796 744 L 3217 744 mt 3212 744 L 2792 587 mt 3217 587 L 2792 942 mt 3217 942 L 3217 942 mt 3217 587 L 2792 942 mt 2792 587 L gs 2792 587 426 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 1 2 0 2 0 2 0 2 1 2 0 2 0 2 1 3 0 2 1 2 1 2 0 2 1 2 1 2 1 2 2 3 1 2 2 2 1 2 2 2 2 2 2 2 3 2 3 3 2 2 4 2 3 2 4 2 4 2 4 2 4 2 5 3 5 2 6 2 5 2 6 2 7 2 6 2 7 2 7 3 7 2 8 2 7 2 8 2 8 2 8 2 8 2 8 3 7 2 8 2 8 2 8 2 7 2 7 2 7 2 6 3 6 2 6 2 5 2 5 2 4 2 3 2 3 2 3 3 2 2 1 2 0 2 0 2 -1 2 -1 2 -2 2 -2 3 -3 2 -4 2 -4 2 -4 2 -5 2 -6 2 -5 2 -6 3 -6 2 -6 2 -6 2 -7 2 -6 2 -6 2 -7 2 -6 3 -6 2 -5 3008 782 100 MP stroke 2 -6 2 -5 2 -5 2 -4 2 -5 2 -3 3 -4 2 -3 2 -2 2 -3 2 -1 2 -2 2 -1 2 0 3 -1 2 0 2 1 2 0 2 1 2 1 2 1 2 1 3 2 2 1 2 2 2 1 2 2 2 1 2 1 2 2 3 1 2 1 2 0 2 1 2 0 2 0 2 0 2 -1 3 -1 2 -1 2 -1 2 -2 2 -2 2 -2 2 -3 2 -2 3 -3 2 -3 2 -4 2 -3 2 -4 2 -3 2 -4 2 -4 3 -4 2 -4 2 -4 2 -4 2 -4 2 -4 2 -4 2 -3 3 -4 2 -4 2 -3 2 -4 2 -3 2 -3 2 -3 2 -3 3 -3 2 -2 2 -3 2 -2 2 -2 2 -2 2 -2 2 -1 3 -2 2 -1 2 -2 2 -1 2 -1 2 -1 2 -1 2 -1 3 -1 2 0 2 -1 2 -1 2 0 2 -1 2 0 2 0 3 -1 2 0 2 0 2 0 2 0 2798 940 100 MP stroke 2 -1 2 0 2 0 2792 941 4 MP stroke gr 6 w 2813 680 mt (\(k\)) s 3 w 1 sg 0 355 425 0 0 -355 3354 942 4 MP PP -425 0 0 355 425 0 0 -355 3354 942 5 MP stroke 2 w DO SO 3 w 0 sg 3354 587 mt 3779 587 L 3354 942 mt 3779 942 L 3779 942 mt 3779 587 L 3354 942 mt 3354 587 L 3354 942 mt 3779 942 L 3354 942 mt 3354 587 L 3354 942 mt 3354 937 L 3354 587 mt 3354 591 L 3566 942 mt 3566 937 L 3566 587 mt 3566 591 L 3779 942 mt 3779 937 L 3779 587 mt 3779 591 L 3354 942 mt 3358 942 L 3779 942 mt 3774 942 L 3354 744 mt 3358 744 L 3779 744 mt 3774 744 L 3354 587 mt 3779 587 L 3354 942 mt 3779 942 L 3779 942 mt 3779 587 L 3354 942 mt 3354 587 L gs 3354 587 426 356 MR c np 6 w 1 0 3 0 2 0 2 1 2 0 2 1 2 0 2 1 2 0 3 1 2 1 2 0 2 1 2 1 2 1 2 1 2 1 3 2 2 1 2 2 2 1 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 3 2 3 2 3 2 3 3 4 2 3 2 4 2 4 2 4 2 4 2 4 2 4 3 5 2 4 2 5 2 5 2 5 2 5 2 5 2 5 3 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 3 5 2 5 2 4 2 5 2 4 2 4 2 4 2 4 3 3 2 3 2 3 2 3 2 3 2 2 2 2 2 1 3 1 2 1 2 1 2 1 2 0 2 0 2 -1 2 -1 3 0 2 -2 2 -1 2 -2 2 -2 2 -2 2 -2 2 -2 3 -3 2 -2 2 -3 2 -3 2 -3 2 -3 2 -3 2 -3 3 -3 2 -3 3570 769 100 MP stroke 2 -3 2 -3 2 -3 2 -3 2 -2 2 -3 3 -2 2 -3 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 3 -1 2 -2 2 -1 2 -1 2 -1 2 -1 2 -1 2 -1 3 -1 2 -1 2 -1 2 -1 2 0 2 -1 2 -1 2 0 3 -1 2 -1 2 -1 2 -1 2 -1 2 -1 2 -1 2 -1 3 -1 2 -2 2 -1 2 -2 2 -1 2 -2 2 -2 2 -2 3 -1 2 -3 2 -2 2 -2 2 -2 2 -3 2 -2 2 -2 3 -3 2 -2 2 -3 2 -3 2 -2 2 -3 2 -3 2 -2 3 -3 2 -2 2 -3 2 -3 2 -2 2 -3 2 -2 2 -2 3 -3 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 2 -2 3 -2 2 -2 2 -1 2 -2 2 -1 2 -2 2 -1 2 -1 3 -1 2 -1 2 -1 2 -1 2 -1 2 -1 2 -1 2 -1 3 0 2 -1 2 -1 2 0 2 -1 3360 937 100 MP stroke 2 0 2 -1 2 0 3354 938 4 MP stroke gr 6 w 3375 680 mt (\(l\)) s 3 w 1 sg 0 355 425 0 0 -355 543 1410 4 MP PP -425 0 0 355 425 0 0 -355 543 1410 5 MP stroke 2 w DO SO 3 w 0 sg 543 1055 mt 968 1055 L 543 1410 mt 968 1410 L 968 1410 mt 968 1055 L 543 1410 mt 543 1055 L 543 1410 mt 968 1410 L 543 1410 mt 543 1055 L 543 1410 mt 543 1405 L 543 1055 mt 543 1059 L 755 1410 mt 755 1405 L 755 1055 mt 755 1059 L 968 1410 mt 968 1405 L 968 1055 mt 968 1059 L 543 1410 mt 547 1410 L 968 1410 mt 963 1410 L 543 1262 mt 547 1262 L 968 1262 mt 963 1262 L 543 1114 mt 547 1114 L 968 1114 mt 963 1114 L 543 1055 mt 968 1055 L 543 1410 mt 968 1410 L 968 1410 mt 968 1055 L 543 1410 mt 543 1055 L gs 543 1055 426 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 1 2 0 2 1 2 1 2 2 2 2 2 2 3 3 2 4 2 5 2 5 2 6 2 7 2 8 2 9 3 9 2 9 2 9 2 9 2 9 2 7 2 6 2 5 3 3 2 0 2 0 2 -3 2 -5 2 -6 2 -7 2 -8 3 -9 2 -8 2 -9 2 -8 2 -7 2 -6 2 -4 2 -3 3 -1 2 1 2 3 2 5 2 7 2 10 2 12 2 14 3 15 2 17 2 18 2 19 2 18 2 17 2 16 2 13 3 11 2 7 2 4 2 0 2 -4 2 -7 2 -11 2 -13 3 -16 2 -17 2 -18 2 -17 2 -18 2 -16 2 -14 2 -13 3 -10 2 -7 759 1345 100 MP stroke 2 -4 2 -2 2 2 2 4 2 7 2 10 3 13 2 14 2 16 2 18 2 17 2 18 2 17 2 16 3 13 2 11 2 7 2 4 2 0 2 -4 2 -7 2 -11 3 -13 2 -16 2 -17 2 -18 2 -19 2 -18 2 -17 2 -15 3 -14 2 -12 2 -10 2 -7 2 -5 2 -3 2 -1 2 1 3 3 2 4 2 6 2 7 2 8 2 9 2 8 2 9 3 8 2 7 2 6 2 5 2 3 2 0 2 0 2 -3 3 -5 2 -6 2 -7 2 -9 2 -9 2 -9 2 -9 2 -9 3 -9 2 -8 2 -7 2 -6 2 -5 2 -5 2 -4 2 -3 3 -2 2 -2 2 -2 2 -1 2 -1 2 0 2 -1 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 549 1409 100 MP stroke 2 0 2 0 2 0 543 1409 4 MP stroke gr 6 w 564 1147 mt (\(m\)) s 3 w 1 sg 0 355 426 0 0 -355 1105 1410 4 MP PP -426 0 0 355 426 0 0 -355 1105 1410 5 MP stroke 2 w DO SO 3 w 0 sg 1105 1055 mt 1531 1055 L 1105 1410 mt 1531 1410 L 1531 1410 mt 1531 1055 L 1105 1410 mt 1105 1055 L 1105 1410 mt 1531 1410 L 1105 1410 mt 1105 1055 L 1105 1410 mt 1105 1405 L 1105 1055 mt 1105 1059 L 1318 1410 mt 1318 1405 L 1318 1055 mt 1318 1059 L 1531 1410 mt 1531 1405 L 1531 1055 mt 1531 1059 L 1105 1410 mt 1109 1410 L 1531 1410 mt 1526 1410 L 1105 1248 mt 1109 1248 L 1531 1248 mt 1526 1248 L 1105 1087 mt 1109 1087 L 1531 1087 mt 1526 1087 L 1105 1055 mt 1531 1055 L 1105 1410 mt 1531 1410 L 1531 1410 mt 1531 1055 L 1105 1410 mt 1105 1055 L gs 1105 1055 427 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 1 2 0 2 1 2 0 3 1 2 1 2 2 2 1 2 2 2 2 2 3 3 3 2 4 2 4 2 4 2 6 2 5 2 6 2 6 3 7 2 6 2 7 2 6 2 7 2 5 2 6 2 4 3 4 2 4 2 2 2 1 2 0 2 0 2 -2 3 -2 2 -3 2 -4 2 -4 2 -4 2 -4 2 -4 2 -3 3 -3 2 -3 2 -2 2 -1 2 -1 2 0 2 1 2 2 3 3 2 3 2 4 2 5 2 6 2 5 2 7 3 6 2 7 2 7 2 8 2 7 2 7 2 8 2 7 3 7 2 6 2 7 2 6 2 6 2 5 2 5 2 5 3 4 2 4 2 3 2 2 2 2 2 1 1322 1182 100 MP stroke 2 1 2 1 3 -1 2 -1 2 -1 2 -2 2 -2 2 -3 2 -4 3 -4 2 -5 2 -5 2 -5 2 -6 2 -6 2 -7 2 -6 3 -7 2 -7 2 -8 2 -7 2 -7 2 -8 2 -7 2 -7 3 -6 2 -7 2 -5 2 -6 2 -5 2 -4 2 -3 3 -3 2 -2 2 -1 2 0 2 1 2 1 2 2 2 3 3 3 2 3 2 4 2 4 2 4 2 4 2 4 2 3 3 2 2 2 2 0 2 0 2 -1 2 -2 2 -4 3 -4 2 -4 2 -6 2 -5 2 -7 2 -6 2 -7 2 -6 3 -7 2 -6 2 -6 2 -5 2 -6 2 -4 2 -4 2 -4 3 -3 2 -3 2 -2 2 -2 2 -1 2 -2 2 -1 3 -1 2 0 2 -1 2 0 2 -1 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 1111 1409 100 MP stroke 2 0 2 0 2 0 1105 1409 4 MP stroke gr 6 w 1126 1145 mt (\(n\)) s 3 w 1 sg 0 355 426 0 0 -355 1667 1410 4 MP PP -426 0 0 355 426 0 0 -355 1667 1410 5 MP stroke 2 w DO SO 3 w 0 sg 1667 1055 mt 2093 1055 L 1667 1410 mt 2093 1410 L 2093 1410 mt 2093 1055 L 1667 1410 mt 1667 1055 L 1667 1410 mt 2093 1410 L 1667 1410 mt 1667 1055 L 1667 1410 mt 1667 1405 L 1667 1055 mt 1667 1059 L 1880 1410 mt 1880 1405 L 1880 1055 mt 1880 1059 L 2093 1410 mt 2093 1405 L 2093 1055 mt 2093 1059 L 1667 1410 mt 1671 1410 L 2093 1410 mt 2088 1410 L 1667 1248 mt 1671 1248 L 2093 1248 mt 2088 1248 L 1667 1087 mt 1671 1087 L 2093 1087 mt 2088 1087 L 1667 1055 mt 2093 1055 L 1667 1410 mt 2093 1410 L 2093 1410 mt 2093 1055 L 1667 1410 mt 1667 1055 L gs 1667 1055 427 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 1 2 0 2 0 2 1 2 1 3 1 2 1 2 1 2 2 2 2 2 3 2 3 2 3 3 4 2 5 2 5 2 5 2 7 2 6 2 8 3 7 2 8 2 8 2 8 2 9 2 7 2 8 2 7 3 7 2 6 2 5 2 5 2 4 2 3 2 3 2 2 3 1 2 1 2 1 2 1 2 0 2 1 2 1 3 1 2 1 2 2 2 2 2 3 2 3 2 3 2 4 3 4 2 4 2 4 2 5 2 4 2 4 2 4 2 4 3 3 2 3 2 3 2 2 2 2 2 1 1884 1181 100 MP stroke 2 1 2 0 3 0 2 -1 2 -1 2 -2 2 -2 2 -3 2 -3 3 -3 2 -4 2 -4 2 -4 2 -4 2 -5 2 -4 2 -4 3 -4 2 -4 2 -3 2 -3 2 -3 2 -2 2 -2 2 -1 3 -1 2 -1 2 -1 2 0 2 -1 2 -1 2 -1 3 -1 2 -2 2 -3 2 -3 2 -4 2 -5 2 -5 2 -6 3 -7 2 -7 2 -8 2 -7 2 -9 2 -8 2 -8 2 -8 3 -7 2 -8 2 -6 2 -7 2 -5 2 -5 2 -5 3 -4 2 -3 2 -3 2 -3 2 -2 2 -2 2 -1 2 -1 3 -1 2 -1 2 -1 2 0 2 0 2 -1 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 1673 1409 100 MP stroke 2 0 2 0 2 0 1667 1409 4 MP stroke gr 6 w 1688 1145 mt (\(o\)) s 3 w 1 sg 0 355 426 0 0 -355 2229 1410 4 MP PP -426 0 0 355 426 0 0 -355 2229 1410 5 MP stroke 2 w DO SO 3 w 0 sg 2229 1055 mt 2655 1055 L 2229 1410 mt 2655 1410 L 2655 1410 mt 2655 1055 L 2229 1410 mt 2229 1055 L 2229 1410 mt 2655 1410 L 2229 1410 mt 2229 1055 L 2229 1410 mt 2229 1405 L 2229 1055 mt 2229 1059 L 2442 1410 mt 2442 1405 L 2442 1055 mt 2442 1059 L 2655 1410 mt 2655 1405 L 2655 1055 mt 2655 1059 L 2229 1410 mt 2233 1410 L 2655 1410 mt 2650 1410 L 2229 1248 mt 2233 1248 L 2655 1248 mt 2650 1248 L 2229 1087 mt 2233 1087 L 2655 1087 mt 2650 1087 L 2229 1055 mt 2655 1055 L 2229 1410 mt 2655 1410 L 2655 1410 mt 2655 1055 L 2229 1410 mt 2229 1055 L gs 2229 1055 427 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 1 2 0 3 0 2 1 2 1 2 1 2 1 2 2 2 2 2 2 3 3 2 3 2 4 2 4 2 5 2 5 2 6 3 7 2 7 2 7 2 8 2 8 2 7 2 8 2 7 3 7 2 6 2 5 2 5 2 3 2 2 2 0 2 0 3 -2 2 -3 2 -4 2 -6 2 -6 2 -6 2 -7 3 -8 2 -7 2 -7 2 -7 2 -7 2 -6 2 -5 2 -4 3 -4 2 -3 2 -1 2 -1 2 1 2 1 2 3 2 4 3 4 2 5 2 6 2 7 2 7 2 7 2 8 3 7 2 7 2 7 2 6 2 5 2 5 2 3 2 3 3 1 2 0 2 0 2 -2 2 -2 2 -3 2 -3 2 -2 3 -3 2 -2 2 -2 2 0 2 1 2 3 2446 1293 100 MP stroke 2 3 2 6 3 7 2 8 2 10 2 11 2 12 2 12 2 13 3 12 2 12 2 10 2 10 2 7 2 6 2 3 2 1 3 -2 2 -4 2 -6 2 -8 2 -11 2 -11 2 -13 2 -14 3 -14 2 -15 2 -14 2 -13 2 -12 2 -12 2 -9 3 -9 2 -6 2 -5 2 -3 2 -2 2 0 2 2 2 2 3 4 2 5 2 5 2 6 2 5 2 6 2 5 2 4 3 4 2 3 2 1 2 1 2 -1 2 -2 2 -3 3 -5 2 -5 2 -6 2 -7 2 -7 2 -8 2 -7 2 -8 3 -8 2 -7 2 -7 2 -7 2 -6 2 -5 2 -5 2 -4 3 -4 2 -3 2 -3 2 -2 2 -2 2 -2 2 -1 3 -1 2 -1 2 -1 2 0 2 0 2 -1 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2235 1409 100 MP stroke 2 0 2 0 2 0 2229 1409 4 MP stroke gr 6 w 2250 1145 mt (\(p\)) s 3 w 1 sg 0 355 425 0 0 -355 2792 1410 4 MP PP -425 0 0 355 425 0 0 -355 2792 1410 5 MP stroke 2 w DO SO 3 w 0 sg 2792 1055 mt 3217 1055 L 2792 1410 mt 3217 1410 L 3217 1410 mt 3217 1055 L 2792 1410 mt 2792 1055 L 2792 1410 mt 3217 1410 L 2792 1410 mt 2792 1055 L 2792 1410 mt 2792 1405 L 2792 1055 mt 2792 1059 L 3004 1410 mt 3004 1405 L 3004 1055 mt 3004 1059 L 3217 1410 mt 3217 1405 L 3217 1055 mt 3217 1059 L 2792 1410 mt 2796 1410 L 3217 1410 mt 3212 1410 L 2792 1248 mt 2796 1248 L 3217 1248 mt 3212 1248 L 2792 1087 mt 2796 1087 L 3217 1087 mt 3212 1087 L 2792 1055 mt 3217 1055 L 2792 1410 mt 3217 1410 L 3217 1410 mt 3217 1055 L 2792 1410 mt 2792 1055 L gs 2792 1055 426 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 1 2 0 3 1 2 0 2 1 2 1 2 1 2 1 2 2 2 1 3 2 2 2 2 3 2 2 2 3 2 3 2 3 2 3 3 3 2 2 2 3 2 3 2 2 2 2 2 2 2 1 3 1 2 1 2 -1 2 0 2 -1 2 -1 2 -1 2 -2 3 -1 2 -2 2 -1 2 -1 2 0 2 0 2 1 2 2 3 3 2 4 2 5 2 6 2 7 2 9 2 10 2 11 3 12 2 13 2 13 2 13 2 14 2 14 2 13 2 13 3 11 2 11 2 9 2 8 2 6 2 5 2 3 2 1 3 1 2 -1 2 -2 2 -3 2 -3 2 -4 2 -4 2 -4 3 -4 2 -4 3008 1181 100 MP stroke 2 -3 2 -3 2 -2 2 -2 2 -2 2 -1 3 -1 2 -1 2 -2 2 -1 2 -2 2 -2 2 -2 2 -3 3 -3 2 -4 2 -5 2 -5 2 -5 2 -6 2 -7 2 -6 3 -7 2 -7 2 -8 2 -7 2 -7 2 -7 2 -6 2 -6 3 -6 2 -5 2 -4 2 -3 2 -3 2 -2 2 -1 2 0 3 0 2 1 2 2 2 2 2 3 2 3 2 3 2 2 3 3 2 2 2 2 2 1 2 0 2 -1 2 -1 2 -2 3 -4 2 -3 2 -5 2 -5 2 -6 2 -5 2 -6 2 -7 3 -6 2 -6 2 -6 2 -5 2 -5 2 -5 2 -4 2 -4 3 -3 2 -3 2 -3 2 -2 2 -1 2 -2 2 -1 2 -1 3 -1 2 -1 2 0 2 0 2 -1 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2798 1409 100 MP stroke 2 0 2 0 2 0 2792 1409 4 MP stroke gr 6 w 2813 1145 mt (\(q\)) s 3 w 1 sg 0 355 425 0 0 -355 3354 1410 4 MP PP -425 0 0 355 425 0 0 -355 3354 1410 5 MP stroke 2 w DO SO 3 w 0 sg 3354 1055 mt 3779 1055 L 3354 1410 mt 3779 1410 L 3779 1410 mt 3779 1055 L 3354 1410 mt 3354 1055 L 3354 1410 mt 3779 1410 L 3354 1410 mt 3354 1055 L 3354 1410 mt 3354 1405 L 3354 1055 mt 3354 1059 L 3566 1410 mt 3566 1405 L 3566 1055 mt 3566 1059 L 3779 1410 mt 3779 1405 L 3779 1055 mt 3779 1059 L 3354 1410 mt 3358 1410 L 3779 1410 mt 3774 1410 L 3354 1248 mt 3358 1248 L 3779 1248 mt 3774 1248 L 3354 1087 mt 3358 1087 L 3779 1087 mt 3774 1087 L 3354 1055 mt 3779 1055 L 3354 1410 mt 3779 1410 L 3779 1410 mt 3779 1055 L 3354 1410 mt 3354 1055 L gs 3354 1055 426 356 MR c np 6 w 1 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 1 2 0 2 0 2 1 2 0 2 1 2 1 2 1 3 2 2 1 2 2 2 3 2 3 2 3 2 5 2 4 3 6 2 6 2 7 2 7 2 9 2 8 2 10 2 10 3 11 2 11 2 11 2 11 2 11 2 11 2 11 2 9 3 10 2 8 2 8 2 7 2 6 2 5 2 4 2 3 3 3 2 2 2 1 2 1 2 1 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 -1 3570 1174 100 MP stroke 2 -1 2 -1 2 -1 2 -1 2 -2 2 -3 3 -3 2 -3 2 -3 2 -4 2 -4 2 -5 2 -5 2 -5 3 -5 2 -6 2 -5 2 -6 2 -5 2 -5 2 -5 2 -5 3 -4 2 -4 2 -4 2 -3 2 -3 2 -2 2 -2 2 -1 3 -1 2 -1 2 -1 2 0 2 0 2 -1 2 0 2 -1 3 -1 2 -2 2 -2 2 -2 2 -3 2 -4 2 -4 2 -4 3 -5 2 -5 2 -6 2 -6 2 -6 2 -6 2 -6 2 -6 3 -6 2 -5 2 -5 2 -5 2 -5 2 -4 2 -4 2 -3 3 -3 2 -3 2 -2 2 -2 2 -2 2 -1 2 -2 2 -1 3 0 2 -1 2 -1 2 0 2 0 2 -1 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 3 0 2 0 2 0 2 0 2 0 3360 1409 100 MP stroke 2 0 2 0 2 0 3354 1409 4 MP stroke gr 6 w 3375 1145 mt (\(r\)) s 3 w end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 126 640 1350 4 v 124 727 4 87 v 180 701 a Fq(pdfs)p 328 727 V 67 w(F-ica)97 b(Jade)77 b(Imax)72 b(Kcca)83 b(Kgv)p 1473 727 V 126 730 1350 4 v 124 817 4 87 v 247 791 a(a)p 328 817 V 155 w(4.4)137 b(3.7)h Fr(1.8)131 b Fq(3.7)119 b(3.0)p 1473 817 V 124 904 V 243 878 a(b)p 328 904 V 155 w(5.8)137 b(4.1)h(3.4)131 b(3.7)119 b Fr(2.9)p 1473 904 V 124 990 V 247 964 a Fq(c)p 328 990 V 155 w(2.3)137 b Fr(1.9)h Fq(2.0)131 b(2.7)119 b(2.4)p 1473 990 V 126 994 1350 4 v 124 1080 4 87 v 243 1054 a(d)p 328 1080 V 155 w(6.4)137 b(6.1)h(6.9)131 b(7.1)119 b Fr(5.7)p 1473 1080 V 124 1167 V 247 1141 a Fq(e)p 328 1167 V 155 w(4.9)137 b(3.9)h(3.2)131 b(1.7)119 b Fr(1.5)p 1473 1167 V 124 1254 V 255 1228 a Fq(f)p 328 1254 V 155 w(3.6)137 b(2.7)h Fr(1.0)131 b Fq(1.7)119 b(1.5)p 1473 1254 V 126 1257 1350 4 v 124 1344 4 87 v 243 1318 a(g)p 328 1344 V 155 w(1.7)137 b(1.4)h Fr(0.5)131 b Fq(1.4)119 b(1.3)p 1473 1344 V 124 1431 V 243 1405 a(h)p 328 1431 V 155 w(5.5)137 b(3.9)h Fr(3.2)131 b Fq(4.3)119 b(3.6)p 1473 1431 V 124 1518 V 259 1491 a(i)p 328 1518 V 155 w(8.7)137 b(7.2)h(6.8)131 b(7.8)119 b Fr(6.5)p 1473 1518 V 126 1521 1350 4 v 124 1608 4 87 v 259 1582 a Fq(j)p 328 1608 V 155 w(6.7)137 b(4.6)100 b(57.6)132 b(1.4)119 b Fr(1.3)p 1473 1608 V 124 1694 V 243 1668 a Fq(k)p 328 1694 V 155 w(5.7)137 b(4.0)h(3.5)131 b(3.2)119 b Fr(2.6)p 1473 1694 V 124 1781 V 259 1755 a Fq(l)p 328 1781 V 118 w(12.1)137 b(7.2)100 b(10.4)132 b(4.8)119 b Fr(4.2)p 1473 1781 V 126 1784 1350 4 v 124 1871 4 87 v 222 1845 a Fq(m)p 328 1871 V 155 w(3.6)137 b Fr(2.9)h Fq(4.2)131 b(6.3)119 b(4.6)p 1473 1871 V 124 1958 V 243 1932 a(n)p 328 1958 V 155 w(5.4)137 b(3.5)100 b(30.6)132 b(7.6)119 b Fr(3.0)p 1473 1958 V 124 2045 V 243 2019 a Fq(o)p 328 2045 V 155 w(4.7)137 b Fr(3.3)h Fq(4.4)131 b(5.1)119 b(4.3)p 1473 2045 V 126 2048 1350 4 v 124 2135 4 87 v 243 2109 a(p)p 328 2135 V 155 w(4.1)137 b(3.1)h(7.4)131 b(3.8)119 b Fr(3.0)p 1473 2135 V 124 2222 V 243 2196 a Fq(q)p 328 2222 V 118 w(22.9)100 b(15.8)g(40.9)132 b(5.1)119 b Fr(3.9)p 1473 2222 V 124 2308 V 255 2282 a Fq(r)p 328 2308 V 155 w(6.6)137 b(4.4)h(4.9)131 b(4.3)119 b Fr(3.6)p 1473 2308 V 126 2312 1350 4 v 124 2398 4 87 v 147 2372 a Fq(mean)p 328 2398 V 127 w(6.4)137 b(4.6)100 b(10.7)132 b(4.2)119 b Fr(3.3)p 1473 2398 V 126 2402 1350 4 v -186 2485 a(Fig)o(.)21 b(1)p Fq(.)32 b(\(T)-6 b(op\))21 b(Source)h(density)h(functions.)32 b(\(Bottom\))22 b(Performance)g(of)-186 2572 y(ICA)f(algorithms)h(for)g Fn(m)27 b Fm(=)f(2)p Fq(.)32 b(The)22 b(best)g(performance)h(in)e(each) i(ro)n(w)f(is)-186 2659 y(indicated)e(in)e(bold)i(font.)-61 2994 y(The)i(results)h(in)g(Figure)g(1)g(\(Bottom\))g(sho)n(w)g(that)g (the)i(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)-186 3081 y(algorithms)i(are)f(competiti)n(v)o(e)h(with)f(current)h(algorithms,)h (and)f(are)g(par)o(-)-186 3168 y(ticularly)21 b(successful)i(at)e (handling)i(asymmetric)e(sources)i(\(see,)f(e.g.,)f(the)-186 3254 y(performance)k(for)e(sources)h(j,)f(l)g(and)h(q\).)36 b(In)23 b(Figure)h(2)f(\(T)-6 b(op\),)24 b(which)g(re-)-186 3341 y(ports)29 b(results)f(for)h(random)g(choices)h(of)e(source)i (distrib)o(utions,)h(we)d(see)-186 3428 y(that)19 b(the)j(K)t Fo(E)t(R)t(N)t(E)t(L)t Fq(I)t(C)t(A)17 b(algorithms)j(perform)g(well)f (for)g(lar)o(ger)g(numbers)-186 3515 y(of)26 b(components.)47 b(Finally)-5 b(,)27 b(in)f(Figure)g(2)g(\(Bottom\),)i(we)e(report)g (the)h(re-)-186 3601 y(sults)d(of)f(an)i(e)o(xperiment)f(in)g(which)g (we)g(added)h(random)g(outliers)e(to)h(the)-186 3688 y(source)g(data.)38 b(W)-6 b(e)23 b(see)h(that)g(our)g(algorithms)g (are)g(particularly)f(resistant)-186 3775 y(to)c(outliers.)476 3970 y Fr(6.)45 b(CONCLUSIONS)-186 4122 y Fq(W)-6 b(e)26 b(ha)o(v)o(e)h(presented)h(tw)o(o)f(no)o(v)o(el,)i(k)o(ernel-based)g (measures)f(of)e(statisti-)-186 4208 y(cal)h(dependence.)52 b(These)27 b(measures)i(can)f(be)g(optimized)g(with)f(respect)-186 4295 y(to)j(a)f(parameter)i(matrix,)h(yielding)e(ne)n(w)g(algorithms)g (for)g(ICA.)f(These)-186 4382 y(algorithms)17 b(are)g(competiti)n(v)o (e)h(with)f(current)g(algorithms,)h(and)f(are)g(partic-)-186 4469 y(ularly)i(notable)h(for)e(their)h(resistance)g(to)g(outliers.)-61 4557 y(Our)28 b(approach)h(to)f(ICA)g(is)g(more)g(\003e)o(xible)g(and)h (more)f(demanding)-186 4643 y(computationally)g(than)f(current)f (algorithms,)j(in)m(v)o(olving)e(a)f(search)h(in)f(a)-186 4730 y(reproducing)16 b(k)o(ernel)f(Hilbert)f(space\227an)i(inner)e (loop)h(which)g(is)f(not)h(present)-186 4817 y(in)21 b(other)g(algorithms.)29 b(But)21 b(the)g(problem)g(of)g(measuring)h (\(and)f(minimiz-)-186 4904 y(ing\))h(departure)h(from)f(independence)j (o)o(v)o(er)d(all)g(possible)h(non-Gaussian)-186 4990 y(source)e(distrib)o(utions)e(is)h(a)f(dif)n(\002cult)h(one,)g(and)g (we)g(feel)f(that)h(the)g(\003e)o(xibil-)-186 5077 y(ity)f(pro)o(vided) h(by)f(our)g(approach)i(is)d(appropriately)j(tar)o(geted.)-61 5165 y(Man)o(y)k(other)g(problems)g(at)f(the)g(intersection)h(of)f (graphical)i(models)-186 5252 y(and)i(nonparametric)h(estimation)e(can) h(also)f(be)h(addressed)h(using)f(these)-186 5339 y(tools.)36 b(In)24 b(particular)m(,)g(in)f(recent)h(w)o(ork)g([4],)g(we)f(ha)o(v)o (e)g(generalized)i(ICA)p 2283 3 1358 4 v 2281 90 4 87 v 2340 64 a Fn(m)126 b(N)p 2652 90 V 80 w Fq(F-ica)42 b(Jade)60 b(Imax)i(Kcca)31 b(Kgv)p 3639 90 V 2283 93 1358 4 v 2281 180 4 87 v 2370 154 a(2)85 b(250)p 2652 180 V 122 w(11)167 b(9)139 b(30)e(7)163 b Fr(5)p 3639 180 V 2281 267 V 2454 241 a Fq(1000)p 2652 267 V 160 w(5)k(4)176 b(7)137 b(3)163 b Fr(2)p 3639 267 V 2283 270 1358 4 v 2281 357 4 87 v 2370 331 a Fq(4)47 b(1000)p 2652 357 V 123 w(18)130 b(13)139 b(25)100 b(12)126 b Fr(11)p 3639 357 V 2281 444 V 2450 418 a Fm(4000)p 2652 444 V 160 w Fq(8)167 b(7)139 b(11)e(6)163 b Fr(4)p 3639 444 V 2283 447 1358 4 v 2281 534 4 87 v 2370 508 a Fq(8)47 b(2000)p 2652 534 V 123 w(26)130 b(22)101 b(123)g(30)126 b Fr(20)p 3639 534 V 2281 621 V 2454 595 a Fq(4000)p 2652 621 V 123 w(18)k(16)139 b(41)100 b(16)163 b Fr(8)p 3639 621 V 2283 624 1358 4 v 2281 711 4 87 v 2333 685 a Fq(16)47 b(4000)p 2652 711 V 123 w(42)130 b(38)101 b(130)g(31)126 b Fr(19)p 3639 711 V 2283 714 1358 4 v 2371 1505 a @beginspecial 159 @llx 304 @lly 457 @urx 478 @ury 1609 @rwi @setspecial %%BeginDocument: outlierfinal.eps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: MATLAB, The Mathworks, Inc. %%Title: C:\KernelICA\NIPSpaper\outlierfinal.eps %%CreationDate: 07/01/2002 13:33:02 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: 1 %%BoundingBox: 159 304 457 478 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /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 /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /ISOLatin1Encoding where {pop /WindowsLatin1Encoding 256 array bdef ISOLatin1Encoding WindowsLatin1Encoding copy pop /.notdef/.notdef/quotesinglbase/florin/quotedblbase/ellipsis/dagger /daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/.notdef/.notdef /.notdef/.notdef/quoteleft/quoteright/quotedblleft/quotedblright/bullet /endash/emdash/tilde/trademark/scaron/guilsinglright/oe/.notdef/.notdef /Ydieresis WindowsLatin1Encoding 128 32 getinterval astore pop} {/WindowsLatin1Encoding StandardEncoding bdef} ifelse /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /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 % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} 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 {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile 3 index 0 eq {/ASCIIHexDecode filter} {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } ifelse exch readstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 159 304 457 478 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 1728 5844 csm 186 103 3576 2084 rc 85 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 3871 2188 rf 6 w 0 1749 2515 0 0 -1749 387 1968 4 MP PP -2515 0 0 1749 2515 0 0 -1749 387 1968 5 MP stroke 4 w DO SO 6 w 0 sg 387 219 mt 2902 219 L 387 1968 mt 2902 1968 L 2902 1968 mt 2902 219 L 387 1968 mt 387 219 L 387 1968 mt 2902 1968 L 387 1968 mt 387 219 L 387 1968 mt 387 1942 L 387 219 mt 387 244 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 354 2113 mt (0) s 910 1968 mt 910 1942 L 910 219 mt 910 244 L 877 2113 mt (5) s 1434 1968 mt 1434 1942 L 1434 219 mt 1434 244 L 1368 2113 mt (10) s 1958 1968 mt 1958 1942 L 1958 219 mt 1958 244 L 1892 2113 mt (15) s 2482 1968 mt 2482 1942 L 2482 219 mt 2482 244 L 2416 2113 mt (20) s 387 1968 mt 412 1968 L 2902 1968 mt 2876 1968 L 286 2012 mt (0) s 387 1676 mt 412 1676 L 2902 1676 mt 2876 1676 L 186 1720 mt (0.1) s 387 1385 mt 412 1385 L 2902 1385 mt 2876 1385 L 186 1429 mt (0.2) s 387 1093 mt 412 1093 L 2902 1093 mt 2876 1093 L 186 1137 mt (0.3) s 387 801 mt 412 801 L 2902 801 mt 2876 801 L 186 845 mt (0.4) s 387 510 mt 412 510 L 2902 510 mt 2876 510 L 186 554 mt (0.5) s 387 219 mt 2902 219 L 387 1968 mt 2902 1968 L 2902 1968 mt 2902 219 L 387 1968 mt 387 219 L gs 387 219 2516 1750 rc DO 1 -1 105 23 105 -37 105 57 105 -33 104 -20 105 36 105 -14 105 28 105 -93 104 8 105 56 105 -8 105 18 105 -93 104 -35 105 63 105 -5 105 -66 105 -64 104 -49 105 -52 105 -127 105 -167 104 -474 387 1848 26 MP stroke gr DO SO 18 18 387 1848 FO 18 18 491 1374 FO 18 18 596 1207 FO 18 18 701 1080 FO 18 18 806 1028 FO 18 18 910 979 FO 18 18 1015 915 FO 18 18 1120 849 FO 18 18 1225 844 FO 18 18 1330 907 FO 18 18 1434 872 FO 18 18 1539 779 FO 18 18 1644 797 FO 18 18 1749 789 FO 18 18 1854 845 FO 18 18 1958 853 FO 18 18 2063 760 FO 18 18 2168 788 FO 18 18 2273 774 FO 18 18 2378 810 FO 18 18 2482 790 FO 18 18 2587 757 FO 18 18 2692 814 FO 18 18 2797 777 FO 18 18 2902 800 FO DO gs 387 219 2516 1750 rc 1 0 105 44 105 20 105 -1 105 14 104 -79 105 80 105 -58 105 64 105 -108 104 10 105 -31 105 61 105 -6 105 -35 104 -70 105 83 105 35 105 -149 105 -1 104 18 105 -86 105 -113 105 -262 104 -412 387 1822 26 MP stroke gr SO 0 j -20 35 -20 -35 40 0 367 1834 4 MP DP -20 35 -20 -35 40 0 471 1422 4 MP DP -20 35 -20 -35 40 0 576 1160 4 MP DP -20 35 -20 -35 40 0 681 1047 4 MP DP -20 35 -20 -35 40 0 786 961 4 MP DP -20 35 -20 -35 40 0 890 979 4 MP DP -20 35 -20 -35 40 0 995 978 4 MP DP -20 35 -20 -35 40 0 1100 829 4 MP DP -20 35 -20 -35 40 0 1205 864 4 MP DP -20 35 -20 -35 40 0 1310 947 4 MP DP -20 35 -20 -35 40 0 1414 877 4 MP DP -20 35 -20 -35 40 0 1519 842 4 MP DP -20 35 -20 -35 40 0 1624 836 4 MP DP -20 35 -20 -35 40 0 1729 897 4 MP DP -20 35 -20 -35 40 0 1834 866 4 MP DP -20 35 -20 -35 40 0 1938 876 4 MP DP -20 35 -20 -35 40 0 2043 768 4 MP DP -20 35 -20 -35 40 0 2148 832 4 MP DP -20 35 -20 -35 40 0 2253 774 4 MP DP -20 35 -20 -35 40 0 2358 854 4 MP DP -20 35 -20 -35 40 0 2462 775 4 MP DP -20 35 -20 -35 40 0 2567 789 4 MP DP -20 35 -20 -35 40 0 2672 788 4 MP DP -20 35 -20 -35 40 0 2777 808 4 MP DP -20 35 -20 -35 40 0 2882 852 4 MP DP DO gs 387 219 2516 1750 rc 1 0 105 147 105 -180 105 39 105 6 104 -60 105 112 105 50 105 -79 105 15 104 -143 105 -37 105 -68 105 125 105 -134 104 -82 105 -38 105 -176 105 88 105 -85 104 -196 105 -47 105 -48 105 -80 104 -185 387 1735 26 MP stroke gr SO 363 1735 mt 411 1735 L 387 1711 mt 387 1759 L 467 1550 mt 515 1550 L 491 1526 mt 491 1574 L 572 1470 mt 620 1470 L 596 1446 mt 596 1494 L 677 1422 mt 725 1422 L 701 1398 mt 701 1446 L 782 1375 mt 830 1375 L 806 1351 mt 806 1399 L 886 1179 mt 934 1179 L 910 1155 mt 910 1203 L 991 1094 mt 1039 1094 L 1015 1070 mt 1015 1118 L 1096 1182 mt 1144 1182 L 1120 1158 mt 1120 1206 L 1201 1006 mt 1249 1006 L 1225 982 mt 1225 1030 L 1306 968 mt 1354 968 L 1330 944 mt 1330 992 L 1410 886 mt 1458 886 L 1434 862 mt 1434 910 L 1515 752 mt 1563 752 L 1539 728 mt 1539 776 L 1620 877 mt 1668 877 L 1644 853 mt 1644 901 L 1725 809 mt 1773 809 L 1749 785 mt 1749 833 L 1830 772 mt 1878 772 L 1854 748 mt 1854 796 L 1934 629 mt 1982 629 L 1958 605 mt 1958 653 L 2039 644 mt 2087 644 L 2063 620 mt 2063 668 L 2144 565 mt 2192 565 L 2168 541 mt 2168 589 L 2249 615 mt 2297 615 L 2273 591 mt 2273 639 L 2354 727 mt 2402 727 L 2378 703 mt 2378 751 L 2458 667 mt 2506 667 L 2482 643 mt 2482 691 L 2563 673 mt 2611 673 L 2587 649 mt 2587 697 L 2668 712 mt 2716 712 L 2692 688 mt 2692 736 L 2773 532 mt 2821 532 L 2797 508 mt 2797 556 L 2878 679 mt 2926 679 L 2902 655 mt 2902 703 L DO gs 387 219 2516 1750 rc SO 1 0 105 -20 105 -22 105 2 105 15 104 -48 105 20 105 -18 105 -31 105 -46 104 -8 105 -19 105 -34 105 -9 105 -14 104 -52 105 -26 105 -35 105 -3 105 -39 104 -5 105 -3 105 -20 105 -1 104 -16 387 1867 26 MP stroke gr SO 362 1842 mt 412 1892 L 412 1842 mt 362 1892 L 466 1826 mt 516 1876 L 516 1826 mt 466 1876 L 571 1825 mt 621 1875 L 621 1825 mt 571 1875 L 676 1805 mt 726 1855 L 726 1805 mt 676 1855 L 781 1802 mt 831 1852 L 831 1802 mt 781 1852 L 885 1797 mt 935 1847 L 935 1797 mt 885 1847 L 990 1758 mt 1040 1808 L 1040 1758 mt 990 1808 L 1095 1755 mt 1145 1805 L 1145 1755 mt 1095 1805 L 1200 1720 mt 1250 1770 L 1250 1720 mt 1200 1770 L 1305 1694 mt 1355 1744 L 1355 1694 mt 1305 1744 L 1409 1642 mt 1459 1692 L 1459 1642 mt 1409 1692 L 1514 1628 mt 1564 1678 L 1564 1628 mt 1514 1678 L 1619 1619 mt 1669 1669 L 1669 1619 mt 1619 1669 L 1724 1585 mt 1774 1635 L 1774 1585 mt 1724 1635 L 1829 1566 mt 1879 1616 L 1879 1566 mt 1829 1616 L 1933 1558 mt 1983 1608 L 1983 1558 mt 1933 1608 L 2038 1512 mt 2088 1562 L 2088 1512 mt 2038 1562 L 2143 1481 mt 2193 1531 L 2193 1481 mt 2143 1531 L 2248 1463 mt 2298 1513 L 2298 1463 mt 2248 1513 L 2353 1483 mt 2403 1533 L 2403 1483 mt 2353 1533 L 2457 1435 mt 2507 1485 L 2507 1435 mt 2457 1485 L 2562 1450 mt 2612 1500 L 2612 1450 mt 2562 1500 L 2667 1452 mt 2717 1502 L 2717 1452 mt 2667 1502 L 2772 1430 mt 2822 1480 L 2822 1430 mt 2772 1480 L 2877 1410 mt 2927 1460 L 2927 1410 mt 2877 1460 L gs 387 219 2516 1750 rc 1 -1 105 -9 105 -22 105 6 105 -5 104 -33 105 -12 105 7 105 -50 105 -53 104 -15 105 -8 105 -31 105 2 105 -29 104 -44 105 -16 105 -12 105 -18 105 -20 104 -11 105 -7 105 -15 105 -4 104 -7 387 1887 26 MP stroke gr 24 W 387 1887 PD 24 W 491 1880 PD 24 W 596 1876 PD 24 W 701 1861 PD 24 W 806 1854 PD 24 W 910 1843 PD 24 W 1015 1823 PD 24 W 1120 1805 PD 24 W 1225 1793 PD 24 W 1330 1777 PD 24 W 1434 1733 PD 24 W 1539 1704 PD 24 W 1644 1706 PD 24 W 1749 1675 PD 24 W 1854 1667 PD 24 W 1958 1652 PD 24 W 2063 1599 PD 24 W 2168 1549 PD 24 W 2273 1556 PD 24 W 2378 1544 PD 24 W 2482 1511 PD 24 W 2587 1506 PD 24 W 2692 1512 PD 24 W 2797 1490 PD 24 W 2902 1481 PD gs 387 219 2516 1750 rc gr 1 sg 0 713 697 0 0 -713 2951 929 4 MP PP -697 0 0 713 697 0 0 -713 2951 929 5 MP stroke 4 w DO SO 6 w 0 sg 2951 216 mt 3648 216 L 2951 929 mt 3648 929 L 3648 929 mt 3648 216 L 2951 929 mt 2951 216 L 2951 929 mt 3648 929 L 2951 929 mt 2951 216 L 2951 216 mt 3648 216 L 2951 929 mt 3648 929 L 3648 929 mt 3648 216 L 2951 929 mt 2951 216 L 3284 338 mt (Jade) s 3284 476 mt (F-ica) s 3284 614 mt (Imax) s 3284 752 mt (Kcca) s 3284 890 mt (Kgv) s gs 2951 216 698 714 rc DO 200 0 3017 298 2 MP stroke SO gs 3080 261 75 75 rc 18 18 3117 298 FO gr DO 200 0 3017 436 2 MP stroke SO gs 3080 399 75 75 rc -20 35 -20 -35 40 0 3097 448 4 MP DP gr DO 200 0 3017 574 2 MP stroke SO gs 3068 525 99 99 rc 3093 574 mt 3141 574 L 3117 550 mt 3117 598 L gr 200 0 3017 712 2 MP stroke gs 3044 639 147 147 rc 3092 687 mt 3142 737 L 3142 687 mt 3092 737 L gr 200 0 3017 851 2 MP stroke gs 3044 778 147 147 rc 24 W 3117 851 PD gr gr 2674 0 394 1975 2 MP stroke 0 -24 -72 24 72 24 0 -24 3068 1975 5 MP PP 0 0 0 -24 -72 24 72 24 0 -24 3068 1975 6 MP stroke 3172 2008 mt (number ) s 3163 2161 mt (of outliers ) s end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1975 1601 a(Fig)o(.)26 b(2)p Fq(.)44 b(\(T)-6 b(op\))26 b(Performance)h(for)f(lar)o(ger)f(number)j(of)e(components)i Fn(m)p Fq(.)1975 1688 y(\(Bottom\))19 b(Performance)h(as)e(a)h (function)h(of)f(the)g(number)h(of)f(outliers.)1975 2033 y(to)k(a)g(model)g(that)g(no)g(longer)h(requires)f(the)g(sources)g(to)g (be)g(independent,)1975 2120 y(b)o(ut)c(requires)h(them)g(only)g(to)f (f)o(actorize)g(according)i(to)e(a)g(tree.)24 b(The)c(depar)o(-)1975 2206 y(ture)g(from)g(a)g(tree)g(distrib)o(ution)g(can)g(be)g(measured)h (in)f(terms)g(of)g(a)f(sum)i(of)1975 2293 y(mutual)f(information)f (terms,)g(and)g(approximated)i(using)e(the)g(KGV)-10 b(.)2660 2498 y Fr(7.)45 b(REFERENCES)2013 2662 y Fq([1])c(S.)29 b(Amari,)i(A.)e(Cichocki,)j(and)e(H.)f(H.)g(Y)-7 b(ang.)60 b(A)29 b(ne)n(w)h(learning)2141 2749 y(algorithm)g(for)f(blind)g (signal)g(separation.)60 b(In)29 b Fp(Adv)-6 b(.)29 b(in)g(NIPS,)f(8)p Fq(,)2141 2835 y(1996.)2013 2955 y([2])41 b(T)-6 b(.)23 b(W)-7 b(.)21 b(Anderson.)40 b Fp(An)22 b(Intr)m(oduction)i(to)e (Multivariate)h(Statistical)2141 3042 y(Analysis)p Fq(.)k(W)m(ile)o(y) 18 b(&)h(Sons,)g(1984.)2013 3162 y([3])41 b(F)-6 b(.)18 b(R.)h(Bach)g(and)h(M.)e(I.)h(Jordan.)28 b(K)n(ernel)19 b(independent)i(component)2141 3249 y(analysis.)28 b Fp(J)n(.)18 b(of)h(Mac)o(hine)h(Learning)g(Resear)m(c)o(h)p Fq(,)f(3:1\22648,)h(2002.)2013 3369 y([4])41 b(F)-6 b(.)30 b(R.)g(Bach)h(and)g(M.)g(I.)f(Jordan.)65 b(T)m(ree-dependent)33 b(component)2141 3455 y(analysis.)28 b(In)19 b Fp(Pr)m(oc.)f(U)m(AI)p Fq(,)g(2002.)2013 3575 y([5])41 b(A.)17 b(J.)f(Bell)g(and)i(T)-6 b(.)16 b(J.)h(Sejno)n(wski.)22 b(An)17 b(information-maximization)2141 3662 y(approach)g(to)d(blind)h(separation)h(and)f(blind)g(decon)m(v)o (olution.)k Fp(Neur)o(al)2141 3749 y(Computation)p Fq(,)h (7\(6\):1129\2261159,)i(1995.)2013 3869 y([6])41 b(P)-8 b(.)15 b(J.)g(Bick)o(el,)h(C.)f(A.)g(J.)g(Klaassen,)h(Y)-10 b(.)15 b(Rito)o(v)-5 b(,)16 b(and)h(J.)e(A.)g(W)-6 b(ellner)l(.)19 b Fp(Ef-)2141 3956 y(\002cient)i(and)g(Adaptive)g(Estimation)f(for)g (Semipar)o(ametric)h(Models)p Fq(.)2141 4042 y(Springer)o(-V)-8 b(erlag,)19 b(1998.)2013 4162 y([7])41 b(J.-F)-6 b(.)20 b(Cardoso.)32 b(High-order)22 b(contrasts)f(for)f(independent)j(compo-) 2141 4249 y(nent)d(analysis.)27 b Fp(Neur)o(al)19 b(Computation)p Fq(,)h(11\(1\):157\226192,)h(1999.)2013 4369 y([8])41 b(A.)23 b(Hyv)t(\250)-29 b(arinen,)25 b(J.)e(Karhunen,)j(and)e(E.)e (Oja.)40 b Fp(Independent)26 b(Com-)2141 4456 y(ponent)20 b(Analysis)p Fq(.)27 b(W)m(ile)o(y)18 b(&)h(Sons,)g(2001.)2013 4576 y([9])41 b(T)-6 b(.-W)f(.)24 b(Lee,)i(M.)f(Girolami,)h(and)g(T)-6 b(.)25 b(J.)g(Sejno)n(wski.)47 b(Independent)2141 4663 y(component)34 b(analysis)e(using)h(an)f(e)o(xtended)h(Infomax)f (algorithm)2141 4749 y(for)23 b(mix)o(ed)g(sub-gaussian)g(and)g(super)o (-gaussian)h(sources.)38 b Fp(Neur)o(al)2141 4836 y(Computation)p Fq(,)20 b(11\(2\):417\226441,)i(1999.)1975 4956 y([10])42 b(B.)20 b(Sch)6 b(\250)-31 b(olk)o(opf)22 b(and)f(A.)f(J.)g(Smola.)31 b Fp(Learning)22 b(with)e(K)m(ernels)p Fq(.)32 b(MIT)2141 5043 y(Press,)19 b(2001.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF