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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mpmpaper_final5.dvi %%CreationDate: Wed Nov 06 21:23:03 2002 %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMBX10 CMCSC10 CMTI10 CMR10 CMBX8 CMSY8 CMR8 %%+ CMMI8 CMBX6 CMSY10 CMMI10 MSBM10 CMEX10 CMMI6 MSAM10 CMR6 Helvetica %%+ CMSY6 CMTT10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: DVIPS.EXE -P pdf mpmpaper_final5 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2002.11.06:2122 %%BeginProcSet: tex.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/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]{Ci}imagemask restore}B/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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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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 161 /minus put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e7a4a832f89e4f9cc 36f6027a64831992e98b1707ddc56e562de9ad0278d89f3416dfab55ea0e5b25 9cfdc4787442890a5a8986bd394cfd87942f66b3712745aa18e52c6bd1d44d48 3f75ca3a5d39391f2bb9441314fbb25ed2b6813e66bbdec80f12a88c2f94f601 fdb6b48613c26c3ac738dcd4ab05a16c34aac07eb789901b095230281fc71f04 5ac83d60c6b82255c6276398134634a8135902a31a52e84b55aece4be46e9d93 72e8c2738a3ae29ce61298d5c2e792bc098a143f3f08614489 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523da5d7776ec0bb452 fc23761a3e881b097d783e662049dba7fa44f1d3ff54ed2ef41b4cf563ce14c8 fbdb4dd65ef24c23ec4cccf2f9f8431a36afc36d56046c2d6ba92627e86ad2b3 e72e2f4dbb44c7b934c8c2ae992f6f88d5e1ae4a7061c905a1e5d8f1163364ac 7a30ffae1ceb30fb09e61f505c90161533407f0514aeb72205f5c6d14c24edee b56041eae6b47743e5320f89680659e4c22cbbb10a2b6e575cf5819488dba627 52da0c74e4ff52b05d827feda0befc7305c3070afc2d044e0e5d37b697e4d645 394ca10346a5b30ec1fda41aa7979e0715b5bf7be69f86dff3a49814c54f0dc5 e96391f692e56d24a3108ecaf5a14b4eeb33fffc95a5874294e1207fea991b5d f5a6805837f53bd137b6eecdb497488cfba763241c240a796b9a5c165f11e5ab 092bcf7861e6a139a6b9f8332a81836710058e070edfa82080377e582c874f0d 3c05857ced765bfb79d05545393fa403f67f6466c9c10c1a168d5ce99a0a8f8c d7dc88c78df5501769328d64492a9f922a49fe75c07e94c1510c0637929a2e0e c9fb2a8d85a59dc3de9c27291b60842489a3cb1186b3cfc34c1dc402044dfed8 be1889239e2b91712bb8730b0600721c99943f6ce5ec30720fbee188e1541745 4dc5f30d1235708c18961d575a1ac35f8355882430308281251e5070018e4339 5656764ae20d2abf977d2c449f848fcf6542a0d371e4acd5307ec167a0e5ab2d c96b0863a4ad3468ec9f58b270cb42d8b1e1d136601813cef26a8719528106de 798fe5964dae628d785b970e541dfbcd5dd9dd313dc4fb1a0a293ace6537c986 e9a6ff7001ea719fd543bc05a953c3bb286b693aae90cd2011717619e9f0a17e 85fbb3a94cb5377a9ec6ee2d07dccc17be6d5ae17c03936e3d5955dd58ced70d fb8da3c964efc094ad7328a29189ac94bc706b89fc54c4044609165137721c8d 4b536ca30d4f912c4b1c430a0b575f967bc42b9fd04a410ec09c69e5207b0b02 4c90d6af9b11e0efe35f9c3cbb8499e9692902c934b16c9c7aeccc52a9c25e42 63ea14647c5fa8573335c6e6c9d95b3120aac860332974ff063d04e953eb453f f019d3488d94e0da44b227f7420295842f06ad4d69234ee56cbda2cf9a939fa5 7d5790f66e7e3fd846fd2b7aea5db0e2ca54f15b4bcffd54d6627052e1e80442 5332f71fc5a9a0ea9c4eedd468b8e77d70407bdb47dc43f6e4fde693d0e7881f 4e7736132827c92022575427b995996bc71139713c85d747a1f72da1b20760ce 44854416 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % 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 (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 161 /squaredot put dup 164 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa4f3118e6b59bd7a2c35420557fc465c b2c852523436519025d745b1cc131ac71f57d1c7f0b69de706e5279271eef6b8 cc29744536b62039b99ccaece0be0fc44567a8d7c88d182471437f6037a2927d 7ffe8d73029e19d3898c569d13d201ebb93d6068c4fea6194e86be7bfec4c79c e30f3cf340770ad95ede586c0eeb09bde1717d6309affb0c6cd70e3c425c41b1 ef567d027e1127a8f289eb652c0223c8a233c3501c39cc6ac8155d3f0ec893a7 6bb507f17ea448fc5ed30875d014502eddf49ae7b200f9109a72f687d7b41d3d c95e1210f1d74f5463a4876759ce7476ea5c635036f1ebe44fcc0e2add9bb762 9e28722245b645f72ea99eccc18d110de365ec80a1a38a0caf07a4f0b8bba6df 3cd834f49ab80e5a1f2b09031b4b11c305267b574dabeb6b92647f27dfb2bba1 bca2c33a1b9a3f90472768c29884bb7e8a20567fe0a057d4dc71e6ca4431c135 759c07722d2ee449bb009246eb2b37c15ef24112ef9d233e764f2ae5f2e4327e 9c8ef6386c150a27f14e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /Gamma put dup 164 /Lambda put dup 39 /phi1 put dup 70 /F put dup 84 /T put dup 105 /i put dup 107 /k put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 939205511ebfbac6b51a838f6e00d92a916021775f12c1af6c898c8f37aef240 5a281e2c955a2ebfc6a181326a41ff4b0b7249387be26e16611887fd2fc0126d c928cadc0a73cb4a49962168c5356b33c78b439889fa5d1e92e43a69338862ac a6174270ee43bd93294245dde9111b4f8de275ed04d0c9c8c1693c31e6d06685 7b73fc333a5fa80aaa82fbe5e89d62e348893b598202ab2ac5db0e657da112ed eff4cbe5c2b4ee2141aae7e1fb8c78bb0783b1a6ca76c113c181e5b0f76c7d1e 7e888cee9821a91bacc8b45391a94ff7dfcbda173fab7a8d9ceb048799293148 c890750a3a63bddbc01b15f493905822ecec3176cef95b08b21e8e6c8cd19f6a 9c1e89959df289ed9dfa881c1bd62880d492404f4a9d803e64d44d332218604b 427b3b8e00dc0656ac5fc2af07443bb66658a7bf7db1e80ff788f42e5c068b4e 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 161 /parenleftbig put dup 162 /parenrightbig put dup 163 /bracketleftbig put dup 164 /bracketrightbig put dup 169 /braceleftbig put dup 170 /bracerightbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 189 /braceleftbigg put dup 195 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 80 /summationtext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put dup 116 /radicalbt put dup 117 /radicalvertex put dup 118 /radicaltp put 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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4fe89149a3dfb4bedefb663d0ce9c13838777d6617443962a15920a4359cdc1d f045a975b7c13b4cd8e473498be642ff0118bd3b30b260f7ac98ea1e9225be06 6255e28f3705b36d2c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 174 /ff put dup 175 /fi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 189 /ae put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H 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 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 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71d4d7c697606318fe2a4af847146f0402041193a0e4a347dce116949401f717 99d89845620dbb7c84b899fe7afd950892f317589211ff2e45db45a9a9d3b3ee 6a11c7fa855b8718417f566257e5d766f3ec56b84fd841fc87f8a58c19e653e8 0e2fbc853808d0a203 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 195 /psi put dup 33 /omega put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 73 /I put dup 75 /K put dup 78 /N put dup 79 /O put dup 88 /X put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 115 /s put dup 116 /t put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 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67154a69b64c85b74d04667c3125 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 169 /circleplus put dup 170 /circleminus put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 195 /arrowleft put dup 33 /arrowright put dup 41 /arrowdblright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 68 /D put dup 69 /E put dup 72 /H put dup 76 /L put dup 78 /N put dup 81 /Q put dup 83 /S put dup 88 /X put dup 89 /Y put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 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379837cc078af6446162214f8d00013a8e86a6de4291e34ec13ff40707ea0d69 63f32d27815022c98e736260b81c7700588b1251c2b4edd100a393765d3b0333 4296b09fa9fd09bce1361a999335edb7327c33f8f793 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX6 %!PS-AdobeFont-1.1: CMBX6 1.0 %%CreationDate: 1991 Aug 20 16:35:30 % 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 (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 120 /x put dup 121 /y put readonly def /FontBBox{-49 -250 1367 753}readonly def /UniqueXX 5000764 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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c5a426ff2dcf10d1565bdad64c206d00ba61f12d793bd36a1bb4c2f37fbb81f0 e13f7cf38e0ce6d4d5c65b242081514513246e14bb89139c8fe6dfa898c48a5b 54f58397fb305750f904575a8e3e10e3c6d850fe949cc23df2664a862bafa19e 393775096e0156fa5ec02e66c00b010db26a2c2503 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % 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 (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 195 /psi put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 70 /F put dup 76 /L put dup 78 /N put dup 83 /S put dup 84 /T put dup 88 /X put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 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037cd420ecf291d763e85495544481a0497372382675cfd38d1d4b44aae02835 d080210f5a713c88b9ba2facfbbd30894c2c6a9b3ed5f801393dc9600f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % 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 (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 169 /Phi put dup 170 /Psi put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress 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 51 /three put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 68 /D put dup 70 /F put dup 77 /M put dup 80 /P put dup 97 /a put dup 98 /b put dup 103 /g put dup 111 /o put dup 114 /r put dup 115 /s put dup 117 /u put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % 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 (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 161 /minus put dup 162 /periodcentered put dup 163 /multiply put dup 164 /asteriskmath put dup 169 /circleplus put dup 170 /circleminus put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 195 /arrowleft put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 78 /N put dup 83 /S put dup 107 /bardbl put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b 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9b7d393b8497aec7cc6bd23b02828ebb7d7bf35cfb1a7244fa3bac0e06636a56 6cf5ff2b075273fb2f02caac406af58d00d63258abaec9e4bf5a7f95da65ed05 b2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 97 /a put dup 99 /c put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-59 -250 1235 750}readonly def 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%%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 34 /quotedblright put dup 35 /numbersign put dup 37 /percent put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J 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 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 73 /I put dup 75 /K put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 196 /dieresis put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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7a8b1190cf9c3f90920a240272887c3ecebba342c3bb86ab3787c63b80934edf 1f5f0cb8b2cb6d4a4c15cc0d2d5440e61fba45200873dcec0dbe51484b67a2ae 59890634e9bd1cddcb74fe772a74f7ac9561c30e7bd7be 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % 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 (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J 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 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 196 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 170 /Psi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 46 /period put dup 64 /at put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put dup 196 /dieresis put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 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38080 y Fn(\244)21278 36949 y Fp(=)23734 34343 y Ff(\263)24458 34627 y(p)p 25670 34627 4250 49 v 1061 x Fv(a)26348 35338 y Fl(T)26348 35988 y Fn(\244)27082 35688 y Fv(\247)28089 35870 y Fo(x)28716 35688 y Fv(a)29394 35870 y Fn(\244)30189 35688 y Fp(+)31401 34654 y Ff(p)p 32613 34654 4265 49 v 1034 x Fv(a)33291 35338 y Fl(T)33291 35988 y Fn(\244)34025 35688 y Fv(\247)35032 35870 y Fo(y)35674 35688 y Fv(a)36352 35870 y Fn(\244)36878 34343 y Ff(\264)37602 34645 y Fm(2)p 22690 36670 16481 49 v 22690 38430 a Fp(1)h(+)24778 37084 y Ff(\263)25502 37368 y(p)p 26714 37368 4250 49 v 1062 x Fv(a)27392 38079 y Fl(T)27392 38729 y Fn(\244)28125 38430 y Fv(\247)29132 38612 y Fo(x)29760 38430 y Fv(a)30438 38612 y Fn(\244)31233 38430 y Fp(+)32445 37396 y Ff(p)p 33657 37396 4265 49 v 1034 x Fv(a)34335 38079 y Fl(T)34335 38729 y Fn(\244)35068 38430 y Fv(\247)36075 38612 y Fo(y)36718 38430 y Fv(a)37396 38612 y Fn(\244)37922 37084 y Ff(\264)38646 37386 y Fm(2)39304 36949 y Fi(:)0 40715 y Fh(If)434 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b(to)g(class)g Fv(y)19 b Fp(.)513 b(This)331 b(algorithm)0 47716 y(for)g(binary)g(classi\257cation)g(is)f (called)h(the)g(Minimax)f(Probabilit)-34 b(y)331 b(Mac)-34 b(hine)332 b(\(MPM\))f(for)g(binary)g(classi\257cation.)1882 50727 y Fv(Pro)39 b(of)454 b(of)g(theorem:)534 b Fp(Consider)396 b(the)g(second)f(constrain)-34 b(t)397 b(in)e(\(1\),)j(with)e(giv)-34 b(en)395 b Fv(a)336 b Fj(6)p Fp(=)g(0.)536 b(Using)396 b(Lemma)e(1,)0 52232 y(this)405 b(constrain)-34 b(t)405 b(can)f(b)34 b(e)404 b(expressed)g(as)11694 55269 y Fi(b)269 b Fj(\241)g Fv(a)14373 54768 y Fl(T)15181 55252 y Fp(\271)15107 55269 y Fv(y)356 b Fj(\270)336 b Fi(\267)p Fp(\()p Fi(\256)t Fp(\))19897 53812 y Ff(q)p 21111 53812 3739 49 v 21111 55269 a Fv(a)21789 54919 y Fl(T)22523 55269 y Fv(\247)23530 55451 y Fo(y)24172 55269 y Fv(a)2424 b Fp(where)1212 b Fi(\267)p Fp(\()p Fi(\256)t Fp(\))339 b(=)35624 53544 y Ff(r)p 36837 53544 3134 49 v 38013 54449 a Fi(\256)p 36970 54990 2868 49 v 36970 56100 a Fp(1)269 b Fj(\241)g 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Fl(\256;)p Fo(a)p Fn(6)p Fm(=0)p Fl(;b)18592 65504 y Fi(\256)1112 b Fp(s.t.)1107 b Fj(\241)p Fi(b)269 b Fp(+)g Fv(a)26831 65004 y Fl(T)27629 65487 y Fp(\271)27565 65504 y Fv(x)336 b Fj(\270)g Fi(\267)p Fp(\()p Fi(\256)t Fp(\))32335 64325 y Ff(p)p 33550 64325 3724 49 v 33550 65504 a Fv(a)34228 65154 y Fl(T)34961 65504 y Fv(\247)35968 65686 y Fo(x)36596 65504 y Fv(a)13177 b Fp(\(8\))23209 68611 y Fi(b)269 b Fj(\241)g Fv(a)25888 68111 y Fl(T)26696 68594 y Fp(\271)26622 68611 y Fv(y)356 b Fj(\270)336 b Fi(\267)p Fp(\()p Fi(\256)t Fp(\))31412 67155 y Ff(q)p 32626 67155 3739 49 v 32626 68611 a Fv(a)33304 68261 y Fl(T)34038 68611 y Fv(\247)35045 68793 y Fo(y)35687 68611 y Fv(a)p Fi(:)25697 72000 y Fp(4)p eop %%Page: 5 5 5 4 bop 0 1218 a Fp(Since)334 b Fi(\267)p Fp(\()p Fi(\256)t Fp(\))i(is)e(a)g(monotone)h(increasing)f(function)h(of)g Fi(\256)t Fp(,)348 b(w)-34 b(e)335 b(can)f(c)-34 b(hange)334 b(v)-67 b(ariables)333 b(and)i(rewrite)f(our)g(problem)0 2723 y(as)10872 4758 y(max)10401 5575 y Fl(\267;)p Fo(a)p Fn(6)p Fm(=0)p Fl(;b)14204 4758 y Fi(\267)1213 b Fp(s.t.)f Fv(a)19628 4257 y Fl(T)20436 4741 y Fp(\271)20361 4758 y Fv(y)289 b Fp(+)269 b Fi(\267)23296 3301 y Ff(q)p 24508 3301 3739 49 v 1457 x Fv(a)25186 4407 y Fl(T)25920 4758 y Fv(\247)26927 4940 y Fo(y)27569 4758 y Fv(a)337 b Fj(\267)f Fi(b)h Fj(\267)f Fv(a)32677 4257 y Fl(T)33476 4741 y Fp(\271)33411 4758 y Fv(x)269 b Fj(\241)g Fi(\267)36326 3578 y Ff(p)p 37539 3578 3724 49 v 37539 4758 a Fv(a)38217 4407 y Fl(T)38950 4758 y Fv(\247)39957 4940 y Fo(x)40585 4758 y Fv(a)o Fi(;)8852 b Fp(\(9\))0 7972 y(the)405 b(optimal)f(v)-67 b(alues)403 b(of)i Fi(\267)f Fp(and)h(of)g(the)f(w)-34 b(orst-case)405 b(misclassi\257cation)f(probabilit)-34 b(y)405 b(1)269 b Fj(\241)g Fi(\256)409 b Fp(b)34 b(eing)404 b(related)g(b)-34 b(y)22584 11736 y Fi(\256)23359 11918 y Fn(\244)24222 11736 y Fp(=)26678 10916 y Fi(\267)27376 10476 y Fm(2)27376 11216 y Fn(\244)p 25634 11457 3312 49 v 25634 12568 a Fp(1)270 b(+)f Fi(\267)28420 12217 y Fm(2)28420 12867 y Fn(\244)29079 11736 y Fi(:)0 15475 y Fp(F)-101 b(urther,)404 b(w)-34 b(e)405 b(can)f(eliminate)g Fi(b)g Fp(from)g(\(9\):)12045 19030 y(max)11906 19847 y Fl(\267;)p Fo(a)p Fn(6)p Fm(=0)15046 19030 y Fi(\267)1107 b Fp(s.t.)g Fv(a)20259 18529 y Fl(T)21067 19013 y Fp(\271)20992 19030 y Fv(y)289 b Fp(+)269 b Fi(\267)23927 17573 y Ff(q)p 25139 17573 3739 49 v 1457 x Fv(a)25817 18680 y Fl(T)26551 19030 y Fv(\247)27558 19212 y Fo(y)28200 19030 y Fv(a)337 b Fj(\267)f Fv(a)31172 18529 y Fl(T)31970 19013 y Fp(\271)31906 19030 y Fv(x)269 b Fj(\241)g Fi(\267)34821 17851 y Ff(p)p 36033 17851 3724 49 v 1179 x Fv(a)36711 18680 y Fl(T)37445 19030 y Fv(\247)38452 19212 y Fo(x)39079 19030 y Fv(a)p Fi(:)9751 b Fp(\(10\))0 22748 y(Since)467 b(w)-34 b(e)468 b(w)-34 b(an)g(t)469 b(to)f(maximize)e Fi(\267)p Fp(,)483 b(the)468 b(inequalities)e(in)i(\(9\))g(will)e(b)34 b(ecome)467 b(equalities)g(at)g(the)h(optim)-34 b(um.)730 b(An)0 24253 y(optimal)404 b(v)-67 b(alue)404 b(of)g Fi(b)g Fp(will)g(th)-34 b(us)406 b(b)34 b(e)404 b(giv)-34 b(en)404 b(b)-34 b(y)13361 27768 y Fi(b)13881 27950 y Fn(\244)14744 27768 y Fp(=)336 b Fv(a)16701 27268 y Fl(T)16701 28068 y Fn(\244)17499 27751 y Fp(\271)17434 27768 y Fv(x)270 b Fj(\241)f Fi(\267)20350 27950 y Fn(\244)20876 26284 y Ff(q)p 22088 26284 4250 49 v 1484 x Fv(a)22766 27418 y Fl(T)22766 28068 y Fn(\244)23499 27768 y Fv(\247)24506 27950 y Fo(x)25134 27768 y Fv(a)25812 27950 y Fn(\244)26674 27768 y Fp(=)337 b Fv(a)28632 27268 y Fl(T)28632 28068 y Fn(\244)29440 27751 y Fp(\271)29365 27768 y Fv(y)288 b Fp(+)270 b Fi(\267)32300 27950 y Fn(\244)32826 26311 y Ff(q)p 34038 26311 4265 49 v 1457 x Fv(a)34716 27418 y Fl(T)34716 28068 y Fn(\244)35449 27768 y Fv(\247)36456 27950 y Fo(y)37099 27768 y Fv(a)37777 27950 y Fn(\244)38302 27768 y Fi(;)0 31052 y Fp(where)376 b Fv(a)4155 31234 y Fn(\244)5057 31052 y Fp(and)h Fi(\267)8084 31234 y Fn(\244)8987 31052 y Fp(are)f(optimal)h(v)-67 b(alues)375 b(of)i Fv(a)f Fp(and)h Fi(\267)g Fp(resp)34 b(ectiv)-34 b(ely)-101 b(.)528 b(Rearranging)376 b(the)h(constrain)-34 b(t)378 b(in)e(\(10\),)383 b(w)-34 b(e)0 32557 y(obtain)405 b(the)g(optimization)f(problem)11429 36249 y(max)11290 37066 y Fl(\267;)p Fo(a)p Fn(6)p Fm(=0)14430 36249 y Fi(\267)1212 b Fp(s.t.)g Fv(a)19853 35749 y Fl(T)20587 36249 y Fp(\()21123 36232 y(\271)21058 36249 y Fv(x)269 b Fj(\241)23350 36232 y Fp(\271)23275 36249 y Fv(y)19 b Fp(\))337 b Fj(\270)g Fi(\267)27018 34540 y Ff(\265)27910 35070 y(p)p 29123 35070 3724 49 v 29123 36249 a Fv(a)29801 35899 y Fl(T)30534 36249 y Fv(\247)31541 36431 y Fo(x)32169 36249 y Fv(a)269 b Fp(+)34328 34792 y Ff(q)p 35540 34792 3739 49 v 1457 x Fv(a)36218 35899 y Fl(T)36951 36249 y Fv(\247)37958 36431 y Fo(y)38601 36249 y Fv(a)39279 34540 y Ff(\266)40373 36249 y Fi(:)9135 b Fp(\(11\))1882 40064 y(If)3251 40047 y(\271)3186 40064 y Fv(x)491 b Fp(=)5921 40047 y(\271)5846 40064 y Fv(y)19 b Fp(,)520 b(then)497 b Fv(a)490 b Fj(6)p Fp(=)h(0)496 b(implies)g Fi(\267)491 b Fp(=)g(0,)519 b(whic)-34 b(h)497 b(in)g(turn)g(yields)f Fi(\256)g Fp(=)490 b(0.)816 b(In)496 b(this)h(case,)519 b(the)497 b(minimax)0 41569 y(probabilit)-34 b(y)433 b(decision)f(problem)g(\(1\))h(do)34 b(es)433 b(not)g(ha)-34 b(v)g(e)432 b(a)h(meaningful)g(solution,)440 b(and)433 b(the)g(optimal)f(w)-34 b(orst-case)0 43075 y(misclassi\257cation)404 b(probabilit)-34 b(y)405 b(is)f(1)269 b Fj(\241)g Fi(\256)19189 43257 y Fn(\244)20052 43075 y Fp(=)337 b(1.)1882 44704 y(Let)593 b(us)i(pro)34 b(ceed)593 b(with)i(the)f(assumption)22553 44687 y(\271)22489 44704 y Fv(x)652 b Fj(6)p Fp(=)25547 44687 y(\271)25473 44704 y Fv(y)19 b Fp(.)1108 b(W)-101 b(e)593 b(observ)-34 b(e)594 b(that)h(condition)f(\(11\))h(is)f(p)34 b(ositiv)-34 b(ely)0 46209 y(homogeneous)301 b(in)e Fv(a)p Fp(:)486 b(if)300 b Fv(a)f Fp(satis\257es)h(\(11\),)321 b Fi(s)p Fv(a)299 b Fp(with)i Fi(s)336 b Fj(\270)h Fp(0)299 b(also)h(do)34 b(es.)503 b(F)-101 b(urthermore,)321 b(\(11\))300 b(implies)f Fv(a)46490 45769 y Fl(T)47224 46209 y Fp(\()47760 46192 y(\271)47695 46209 y Fv(x)60 b Fj(\241)49569 46192 y Fp(\271)49494 46209 y Fv(y)19 b Fp(\))337 b Fj(\270)0 47715 y Fp(0.)695 b(Since)4920 47698 y(\271)4855 47715 y Fv(x)423 b Fj(6)p Fp(=)7455 47698 y(\271)7380 47715 y Fv(y)19 b Fp(,)469 b(w)-34 b(e)457 b(can)f(set)h Fv(a)15675 47275 y Fl(T)16408 47715 y Fp(\()16944 47698 y(\271)16879 47715 y Fv(x)304 b Fj(\241)19240 47698 y Fp(\271)19166 47715 y Fv(y)19 b Fp(\))424 b(=)f(1)456 b(without)i(loss)e(of)h (generalit)-34 b(y)-101 b(.)694 b(This)457 b(implies)f Fv(a)423 b Fj(6)p Fp(=)g(0,)469 b(and)457 b(in)0 49345 y(turn,)3034 48225 y Ff(p)p 4246 48225 3724 49 v 1120 x Fv(a)4924 48995 y Fl(T)5657 49345 y Fv(\247)6664 49527 y Fo(x)7292 49345 y Fv(a)269 b Fp(+)9451 48311 y Ff(p)p 10663 48311 3739 49 v 1034 x Fv(a)11341 48995 y Fl(T)12075 49345 y Fv(\247)13082 49527 y Fo(y)13724 49345 y Fv(a)336 b Fj(6)p Fp(=)h(0.)538 b(Th)-34 b(us)406 b(w)-34 b(e)405 b(can)f(write)g(the)g(optimization)h(problem)f(as)13838 53394 y(max)14300 54121 y Fl(\267;)p Fo(a)16700 53394 y Fi(\267)1107 b Fp(s.t.)g Fi(\267)337 b Fj(\267)23550 51684 y Ff(\265)24442 52214 y(p)p 25654 52214 3724 49 v 1180 x Fv(a)26332 53043 y Fl(T)27066 53394 y Fv(\247)28073 53576 y Fo(x)28700 53394 y Fv(a)269 b Fp(+)30859 51937 y Ff(q)p 32071 51937 3739 49 v 1457 x Fv(a)32749 53043 y Fl(T)33483 53394 y Fv(\247)34490 53576 y Fo(y)35132 53394 y Fv(a)35810 51684 y Ff(\266)36702 51986 y Fn(\241)p Fm(1)21235 56093 y Fv(a)21913 55593 y Fl(T)22647 56093 y Fp(\()23183 56076 y(\271)23118 56093 y Fv(x)g Fj(\241)25410 56076 y Fp(\271)25335 56093 y Fv(y)19 b Fp(\))337 b(=)g(1)p Fi(;)0 59187 y Fp(whic)-34 b(h)405 b(allo)-34 b(ws)404 b(us)h(to)f(eliminate)g Fi(\267)p Fp(:)12763 62280 y(min)13509 63007 y Fo(a)14985 62280 y Fj(k)p Fv(\247)16598 62462 y Fo(x)17226 61779 y Fm(1)p Fl(=)p Fm(2)18692 62280 y Fv(a)p Fj(k)19976 62462 y Fm(2)20771 62280 y Fp(+)269 b Fj(k)p Fv(\247)23596 62462 y Fo(y)24239 61779 y Fm(1)p Fl(=)p Fm(2)25706 62280 y Fv(a)p Fj(k)26990 62462 y Fm(2)28622 62280 y Fp(s.t.)1107 b Fv(a)32030 61779 y Fl(T)32763 62280 y Fp(\()33299 62263 y(\271)33234 62280 y Fv(x)270 b Fj(\241)35527 62263 y Fp(\271)35452 62280 y Fv(y)19 b Fp(\))337 b(=)g(1)p Fi(:)10607 b Fp(\(12\))0 65668 y(The)464 b(ab)34 b(o)-34 b(v)g(e)465 b(problem)f(is)f(con)-34 b(v)g(ex.)718 b(Moreo)-34 b(v)g(er,)478 b(the)465 b(ob)67 b(jectiv)-34 b(e)464 b(function)h(is)f(strictly)f(con)-34 b(v)g(ex)464 b(if)g(either)g Fv(\247)49828 65850 y Fo(x)50919 65668 y Fp(or)0 67174 y Fv(\247)1007 67356 y Fo(y)2045 67174 y Fp(is)395 b(p)34 b(ositiv)-34 b(e)394 b(de\257nite,)k(whic)-34 b(h)396 b(allo)-34 b(ws)395 b(us)h(to)f(conclude)g(the)h(uniqueness)f (of)h(optimal)f Fv(a)g Fp(and)h Fi(b)f Fp(in)g(this)h(case.)0 68679 y(This)405 b(ends)f(our)h(pro)34 b(of)404 b(of)g(Theorem)h(2.)538 b Fd(\244)25697 72000 y Fp(5)p eop %%Page: 6 6 6 5 bop 0 1218 a Fv(2.3)697 b(Solving)466 b(the)f(Optimization)g (Problem)0 3391 y Fp(Problem)323 b(\(6\))i(is)e(a)g(con)-34 b(v)g(ex)324 b(optimization)g(problem,)340 b(more)323 b(precisely)f(a)h(second)h(order)g(cone)f(program)h(\(SOCP\))0 4896 y(\(Bo)-34 b(yd)546 b(and)h(V)-101 b(anden)-34 b(b)34 b(erghe,)546 b(2001\).)964 b(General-purp)34 b(ose)545 b(programs)h(suc)-34 b(h)547 b(as)e(SeDuMi)h(\(Sturm,)h(1999\))f(or)0 6402 y(Mosek)564 b(\(Andersen)h(and)f(Andersen,)h(2000\))f(can)h (handle)f(those)h(problems)f(e\261cien)-34 b(tly)-101 b(.)1018 b(These)564 b(co)34 b(des)564 b(use)0 7907 y(in)-34 b(terior-p)34 b(oin)-34 b(t)400 b(metho)34 b(ds)401 b(for)f(SOCP)g (\(Nestero)-34 b(v)399 b(and)i(Nemiro)-34 b(vsky,)398 b(1994;)i(Lob)34 b(o)399 b(et)h(al.,)e(1998\),)j(whic)-34 b(h)401 b(yield)0 9413 y(a)285 b(w)-34 b(orst-case)286 b(complexit)-34 b(y)285 b(of)g Fi(O)34 b Fp(\()p Fi(n)16076 8973 y Fm(3)16601 9413 y Fp(\).)499 b(Of)286 b(course,)308 b(the)286 b(\257rst)f(and)h(second)g(momen)-34 b(ts)286 b(of)g Fv(x)p Fi(;)202 b Fv(y)303 b Fp(m)-34 b(ust)287 b(b)34 b(e)284 b(estimated)0 10918 y(b)34 b(eforehand,)395 b(using)d(for)f(example)g(sample)h(momen)-34 b(t)392 b(plug-in)h(estimates)34512 10901 y(^)34447 10918 y Fv(x)p Fi(;)35796 10901 y Fp(^)35722 10918 y Fv(y)19 b Fi(;)37216 10608 y Fp(^)37016 10918 y Fv(\247)38023 11100 y Fo(x)38650 10918 y Fi(;)39389 10608 y Fp(^)39189 10918 y Fv(\247)40196 11100 y Fo(y)41230 10918 y Fp(for)43137 10901 y(\271)43073 10918 y Fv(x)o Fi(;)44421 10901 y Fp(\271)44347 10918 y Fv(y)g Fi(;)202 b Fv(\247)46648 11100 y Fo(x)47275 10918 y Fp(,)394 b(and)e Fv(\247)51357 11100 y Fo(y)0 12424 y Fp(resp)34 b(ectiv)-34 b(ely)-101 b(.)839 b(This)505 b(brings)g(the)h(total)f(complexit)-34 b(y)504 b(to)i Fi(O)34 b Fp(\()p Fi(n)29085 11984 y Fm(3)29946 12424 y Fp(+)336 b Fi(N)132 b(n)33059 11984 y Fm(2)33585 12424 y Fp(\),)530 b(where)504 b Fi(N)637 b Fp(is)505 b(the)g(n)-34 b(um)g(b)34 b(er)506 b(of)f(data)0 13929 y(p)34 b(oin)-34 b(ts.)635 b(This)437 b(is)f(the)g(same)g(complexit)-34 b(y)436 b(as)g(the)h(quadratic)f(programs)g(one)h(has)f(to)h(solv)-34 b(e)435 b(in)h(linear)g(supp)34 b(ort)0 15435 y(v)-34 b(ector)404 b(mac)-34 b(hines)404 b(\(Sc)-34 b(h\304)-606 b(olk)-34 b(opf)406 b(and)f(Smola,)f(2002\).)1882 16968 y(T)-101 b(o)390 b(gain)g(more)g(insigh)-34 b(t)390 b(in)-34 b(to)391 b(the)f(nature)h(of)f(the)g(problem,)j(w)-34 b(e)390 b(used)g(the)h(follo)-34 b(wing)390 b(simple)g(and)g(p)34 b(erhaps)0 18473 y(more)404 b(transparen)-34 b(t)405 b(metho)34 b(d)405 b(to)g(globally)e(solv)-34 b(e)404 b(the)g(problem.)1882 20006 y(Notice)304 b(that)i(w)-34 b(e)305 b(can)g(write)g Fv(a)336 b Fp(=)g Fv(a)17895 20188 y Fm(0)18492 20006 y Fp(+)71 b Fv(F)-116 b(u)p Fp(,)323 b(where)305 b Fv(u)337 b Fj(2)g Fg(R)28238 19567 y Fl(n)p Fn(\241)p Fm(1)30067 20006 y Fp(,)324 b Fv(a)31406 20188 y Fm(0)32268 20006 y Fp(=)337 b(\()34084 19989 y(\271)34019 20006 y Fv(x)71 b Fj(\241)35914 19989 y Fp(\271)35840 20006 y Fv(y)18 b Fp(\))p Fi(=)p Fj(k)38342 19989 y Fp(\271)38277 20006 y Fv(x)71 b Fj(\241)40172 19989 y Fp(\271)40098 20006 y Fv(y)18 b Fj(k)41458 19567 y Fm(2)41458 20331 y(2)41984 20006 y Fp(,)324 b(and)306 b Fv(F)337 b Fj(2)f Fg(R)48136 19567 y Fl(n)p Fn(\243)p Fm(\()p Fl(n)p Fn(\241)p Fm(1\))0 21512 y Fp(is)426 b(an)g(orthogonal)h (matrix)f(whose)h(columns)f(span)h(the)g(subspace)g(of)f(v)-34 b(ectors)426 b(orthogonal)h(to)44587 21495 y(\271)44522 21512 y Fv(x)283 b Fj(\241)46843 21495 y Fp(\271)46768 21512 y Fv(y)19 b Fp(.)605 b(Hence,)0 23017 y(w)-34 b(e)405 b(can)f(eliminate)f(the)i(constrain)-34 b(t)405 b(in)f(\(12\))h(and)g (write)f(this)h(as)f(an)g(unconstrained)i(SOCP:)12863 25824 y(min)13572 26551 y Fo(u)15085 25824 y Fj(k)p Fv(\247)16698 26006 y Fo(x)17326 25323 y Fm(1)p Fl(=)p Fm(2)18793 25824 y Fp(\()p Fv(a)19942 26006 y Fm(0)20737 25824 y Fp(+)269 b Fv(F)-116 b(u)p Fp(\))p Fj(k)24561 26006 y Fm(2)25357 25824 y Fp(+)269 b Fj(k)p Fv(\247)28182 26006 y Fo(y)28825 25323 y Fm(1)p Fl(=)p Fm(2)30292 25824 y Fp(\()p Fv(a)31441 26006 y Fm(0)32236 25824 y Fp(+)g Fv(F)-116 b(u)p Fp(\))p Fj(k)36060 26006 y Fm(2)36586 25824 y Fi(:)0 28925 y Fp(An)405 b(equiv)-67 b(alen)-34 b(t)403 b(form)h(is)9590 32174 y(inf)8023 32991 y Fo(u)p Fl(;\257)46 b(>)p Fm(0)p Fl(;\264)32 b(>)p Fm(0)13641 32174 y Fi(\257)333 b Fp(+)16076 31354 y(1)p 16005 31895 750 49 v 16005 33005 a Fi(\257)16887 32174 y Fj(k)p Fv(\247)18500 32356 y Fo(x)19128 31673 y Fm(1)p Fl(=)p Fm(2)20595 32174 y Fp(\()p Fv(a)21744 32356 y Fm(0)22539 32174 y Fp(+)269 b Fv(F)-116 b(u)p Fp(\))p Fj(k)26363 31673 y Fm(2)26363 32473 y(2)27159 32174 y Fp(+)269 b Fi(\264)313 b Fp(+)30650 31354 y(1)p 30631 31895 646 49 v 30631 33005 a Fi(\264)31409 32174 y Fj(k)p Fv(\247)33022 32356 y Fo(y)33664 31673 y Fm(1)p Fl(=)p Fm(2)35131 32174 y Fp(\()p Fv(a)36280 32356 y Fm(0)37075 32174 y Fp(+)269 b Fv(F)-116 b(u)p Fp(\))p Fj(k)40899 31673 y Fm(2)40899 32473 y(2)41426 32174 y Fi(:)8082 b Fp(\(13\))0 35605 y(This)411 b(equiv)-67 b(alence)408 b(can)i(b)34 b(e)409 b(understo)34 b(o)g(d)411 b(b)-34 b(y)411 b(\257xing)f Fv(u)g Fp(and)h(minimizing)e(the)i(ob)67 b(jectiv)-34 b(e)410 b(in)g(\(13\))h(with)g(resp)34 b(ect)0 37110 y(to)411 b Fi(\257)475 b Fp(and)411 b Fi(\264)43 b Fp(.)559 b(Notice)411 b(that)h(a)f(factor)17626 36633 y Fm(1)p 17626 36831 471 49 v 17626 37528 a(2)18640 37110 y Fp(o)-34 b(v)g(er)410 b(eac)-34 b(h)411 b(term)g(has)g(b)34 b(een)411 b(dropp)34 b(ed)412 b(in)f(\(13\).)559 b(This)412 b(is)e(irrelev)-67 b(an)-34 b(t)410 b(for)0 38616 y(the)405 b(optimal)g(v)-67 b(alue)404 b(of)i Fv(u)p Fp(,)f Fi(\257)468 b Fp(and)406 b Fi(\264)43 b Fp(,)405 b(although)h(it)f(has)g(to)g(b)34 b(e)405 b(tak)-34 b(en)405 b(in)-34 b(to)406 b(accoun)-34 b(t)406 b(when)f(computing)h Fi(\267)49116 38798 y Fn(\244)50047 38616 y Fp(and)0 40121 y Fi(\256)775 40303 y Fn(\244)1301 40121 y Fp(.)1882 41654 y(The)416 b(function)i(to)e(b)34 b(e)416 b(minimized)f(is)h(join)-34 b(tly)417 b(con)-34 b(v)g(ex)416 b(in)g Fv(u)p Fp(,)j Fi(\257)64 b Fp(,)418 b Fi(\264)459 b Fp(\(as)417 b(follo)-34 b(ws)416 b(easily)f(from)h(the) h(con)-34 b(v)g(exit)g(y)0 43160 y(of)482 b(the)h(function)g Fi(f)130 b Fp(\()p Fi(x;)202 b(t)p Fp(\))467 b(=)f Fi(x)14353 42720 y Fm(2)14879 43160 y Fi(=t)481 b Fp(o)-34 b(v)g(er)482 b(the)g(domain)h Fj(f)p Fp(\()p Fi(x;)202 b(t)p Fp(\))948 b(:)g Fi(x)466 b Fj(2)h Fg(R)p Fi(;)668 b(t)466 b(>)g Fp(0)p Fj(g)p Fp(\).)772 b(T)-101 b(o)483 b(minimize)e(\(13\),)502 b(w)-34 b(e)0 44665 y(used)362 b(an)g(iterativ)-34 b(e)361 b(least-squares)g(approac)-34 b(h)363 b(based)f(on)g(the)g(ab)34 b(o)-34 b(v)g(e)361 b(form)-34 b(ulation,)371 b(with)363 b(regularization)e(of)g(the)0 46171 y(Hessian)495 b(for)h (computational)h(stabilit)-34 b(y)-101 b(.)812 b(The)496 b(algorithm,)518 b(whic)-34 b(h)496 b(is)f(presen)-34 b(ted)497 b(in)e(high-lev)-34 b(el)495 b(pseudo)34 b(co)g(de)0 47676 y(in)597 b(T)-101 b(able)597 b(1,)645 b(go)34 b(es)596 b(as)h(follo)-34 b(ws.)1117 b(A)-34 b(t)598 b(iteration)e Fi(k)38 b Fp(,)645 b(w)-34 b(e)597 b(\257rst)h(minimize)e(with)h(resp) 34 b(ect)597 b(to)g Fi(\257)64 b(;)202 b(\264)639 b Fp(b)-34 b(y)597 b(setting)0 49182 y Fi(\257)686 49379 y Fl(k)1650 49182 y Fp(=)396 b Fj(k)p Fv(\247)4602 49364 y Fo(x)5229 48701 y Fm(1)p Fl(=)p Fm(2)6696 49182 y Fp(\()p Fv(a)7845 49364 y Fm(0)8664 49182 y Fp(+)293 b Fv(F)-116 b(u)11435 49379 y Fl(k)24 b Fn(\241)p Fm(1)13206 49182 y Fp(\))p Fj(k)14283 49364 y Fm(2)15249 49182 y Fp(\()p Fi(\257)64 b Fp(-step\))441 b(and)f Fi(\264)22941 49379 y Fl(k)23905 49182 y Fp(=)396 b Fj(k)p Fv(\247)26857 49364 y Fo(y)27500 48701 y Fm(1)p Fl(=)p Fm(2)28966 49182 y Fp(\()p Fv(a)30115 49364 y Fm(0)30934 49182 y Fp(+)293 b Fv(F)-116 b(u)33705 49379 y Fl(k)24 b Fn(\241)p Fm(1)35476 49182 y Fp(\))p Fj(k)36553 49364 y Fm(2)37519 49182 y Fp(\()p Fi(\264)43 b Fp(-step\).)647 b(Those)440 b(minimizers)0 50687 y(are)531 b(unique.)920 b(Then)532 b(w)-34 b(e)532 b(minimize)e(with)j(resp)34 b(ect)530 b(to)i Fv(u)g Fp(b)-34 b(y)531 b(solving)g(a)h(least-squares) f(problem)g(in)g Fv(u)h Fp(with)0 52193 y Fi(\257)64 b(;)202 b(\264)515 b Fp(\257xed)472 b(\()p Fv(u)p Fp(-step\).)744 b(As)472 b(already)f(men)-34 b(tioned,)490 b(this)473 b(least-squares)f(step)g(is)g(regularized:)673 b(a)473 b(regularization)0 53698 y(term)544 b Fi(\261)46 b Fv(I)543 b Fp(\(with)j Fi(\261)616 b(>)570 b Fp(0)544 b(small\))g(is)g(added)h (to)g(the)f(Hessian)h(of)f(the)h(least-squares)f(step.)959 b(This)545 b(mak)-34 b(es)544 b(the)0 55204 y(Hessian)431 b(p)34 b(ositiv)-34 b(e)430 b(de\257nite.)618 b(Th)-34 b(us,)439 b(for)430 b(the)h Fv(u)p Fp(-step)h(also,)k(the)431 b(minim)-34 b(um)431 b(is)g(uniquely)f(attained.)619 b(As)430 b(sho)-34 b(wn)0 56709 y(b)g(y)479 b(Bertsek)-67 b(as)478 b(\(1999\),)499 b(these)479 b(features)h(imply)e(the)i(con)-34 b(v)g(ergence)478 b(of)i(the)f(ab)34 b(o)-34 b(v)g(e)480 b(blo)34 b(c)-34 b(k)478 b(co)34 b(ordinate)479 b(descen)-34 b(t)0 58215 y(metho)34 b(d:)0 60855 y Fv(Theorem)464 b(3)h(\(Con)-39 b(v)g(ergence)464 b(of)h(Blo)39 b(c)-39 b(k)464 b(Co)39 b(ordinate)464 b(Descen)-39 b(t)465 b(\(Bertsek)-77 b(as,)463 b(1999\)\))606 b Fh(Supp)-62 b(ose)307 b(that)0 62360 y Fi(f)581 b Fh(is)450 b(c)-62 b(ontinuously)447 b(di\256er)-62 b(entiable)448 b(over)h(a)h(close)-62 b(d)448 b(c)-62 b(onvex)450 b(set)f Fi(X)462 b Fp(=)367 b Fi(X)33684 62542 y Fm(1)34491 62360 y Fj(\243)282 b Fi(X)36720 62542 y Fm(2)37527 62360 y Fj(\243)f Fi(:)202 b(:)g(:)281 b Fj(\243)g Fi(X)42675 62542 y Fl(m)44013 62360 y Fh(\(assume)449 b(that)f Fv(x)0 63866 y Fh(is)475 b(p)-62 b(artitione)g(d)472 b(as)j Fv(x)412 b Fp(=)g(\()p Fv(x)12772 64048 y Fm(1)13297 63866 y Fi(;)202 b Fv(x)14572 64048 y Fm(2)15098 63866 y Fi(;)g(:)g(:)g(:)f(;)h Fv(x)18528 64048 y Fl(m)19415 63866 y Fp(\))475 b Fh(and)g Fv(x)412 b Fj(2)g Fi(X)508 b Fj(,)412 b Fv(x)28997 64048 y Fl(i)29785 63866 y Fj(2)g Fi(X)32009 64048 y Fl(i)32385 63866 y Fi(;)677 b(i)412 b Fp(=)g(1)p Fi(;)202 b(:)g(:)g(:)f(;)h(m)475 b Fh(with)g Fi(X)44102 64048 y Fl(i)44953 63866 y Fh(close)-62 b(d)473 b(c)-62 b(onvex)0 65371 y(sets\).)555 b(F)-93 b(urthermor)-62 b(e,)432 b(supp)-62 b(ose)431 b(that)h(for)h(e)-62 b(ach)432 b Fi(i)h Fh(and)g Fv(x)337 b Fj(2)f Fi(X)95 b Fh(,)434 b(the)e(minimum)16614 68178 y Fp(min)16366 68995 y Fl(\273)16780 69130 y Fe(i)17130 68995 y Fn(2)p Fl(X)18531 69130 y Fe(i)19085 68178 y Fi(f)130 b Fp(\()p Fv(x)21015 68360 y Fm(1)21542 68178 y Fi(;)202 b(:)g(:)g(:)f(;)h Fv(x)24972 68360 y Fl(i)p Fn(\241)p Fm(1)26549 68178 y Fi(;)g(\273)27618 68360 y Fl(i)27994 68178 y Fi(;)g Fv(x)29269 68360 y Fl(i)p Fm(+1)30846 68178 y Fi(;)g(:)g(:)g(:)f(;)h Fv(x)34276 68360 y Fl(m)35163 68178 y Fp(\))25697 72000 y(6)p eop %%Page: 7 7 7 6 bop 2056 6663 47889 45 v 2720 9320 a Fp(Get)404 b(estimates)18955 9303 y(^)18890 9320 y Fv(x)p Fi(;)20239 9303 y Fp(^)20165 9320 y Fv(y)18 b Fi(;)21659 9010 y Fp(^)21458 9320 y Fv(\247)22465 9502 y Fo(x)23093 9320 y Fi(;)23832 9010 y Fp(^)23632 9320 y Fv(\247)24639 9502 y Fo(y)2720 12331 y Fp(Compute)11323 b Fv(a)19568 12513 y Fm(0)20430 12331 y Fj(\303)337 b Fp(\()22515 12314 y(^)22450 12331 y Fv(x)269 b Fj(\241)24742 12314 y Fp(^)24667 12331 y Fv(y)20 b Fp(\))p Fi(=)p Fj(k)27171 12314 y Fp(^)27106 12331 y Fv(x)269 b Fj(\241)29398 12314 y Fp(^)29323 12331 y Fv(y)19 b Fj(k)30684 11891 y Fm(2)30684 12655 y(2)18890 13886 y Fv(F)337 b Fj(2)g Fg(R)22124 13446 y Fl(n)p Fn(\243)p Fm(\()p Fl(n)p Fn(\241)p Fm(1\))26391 13886 y Fp(\(an)405 b(orthogonal)g(matrix)f(whose)h(columns)f(span)18890 15392 y(the)h(subspace)g(of)f(v)-34 b(ectors)404 b(orthogonal)h(to) 39150 15375 y(^)39085 15392 y Fv(x)269 b Fj(\241)41377 15375 y Fp(^)41302 15392 y Fv(y)20 b Fp(\))18890 16995 y Fv(G)337 b Fj(\303)f Fv(F)22748 16555 y Fl(T)23683 16685 y Fp(^)23482 16995 y Fv(\247)24489 17177 y Fo(x)25117 16995 y Fv(F)18890 18598 y(H)h Fj(\303)f Fv(F)22743 18158 y Fl(T)23678 18288 y Fp(^)23477 18598 y Fv(\247)24484 18780 y Fo(y)25127 18598 y Fv(F)18890 20201 y(g)356 b Fj(\303)337 b Fv(F)22369 19761 y Fl(T)23303 19891 y Fp(^)23103 20201 y Fv(\247)24110 20383 y Fo(x)24737 20201 y Fv(a)25415 20383 y Fm(0)18890 21804 y Fv(h)g Fj(\303)g Fv(F)22427 21364 y Fl(T)23361 21495 y Fp(^)23161 21804 y Fv(\247)24168 21986 y Fo(y)24811 21804 y Fv(a)25489 21986 y Fm(0)2720 24815 y Fp(Initialize)11556 b Fi(\257)19576 24997 y Fm(1)20438 24815 y Fp(=)337 b(1,)403 b Fi(\264)23666 24997 y Fm(1)24529 24815 y Fp(=)336 b(1,)404 b Fi(k)375 b Fp(=)336 b(1)2720 27826 y(Rep)34 b(eat)12416 b Fv(M)20213 28014 y Fl(LS)21867 27826 y Fj(\303)23831 27349 y Fm(1)p 23549 27547 1037 49 v 23549 28244 a Fl(\257)24077 28400 y Fe(k)24718 27826 y Fv(G)269 b Fp(+)27683 27349 y Fm(1)p 27428 27547 981 49 v 27428 28244 a Fl(\264)27900 28400 y Fe(k)28541 27826 y Fv(H)g Fp(+)g Fi(\261)46 b Fv(I)18890 29810 y(b)19664 29998 y Fl(LS)21318 29810 y Fj(\303)337 b(\241)24012 28465 y Ff(\263)25151 29333 y Fm(1)p 24868 29531 1037 49 v 24868 30228 a Fl(\257)25396 30384 y Fe(k)26037 29810 y Fv(g)289 b Fp(+)28623 29333 y Fm(1)p 28368 29531 981 49 v 28368 30228 a Fl(\264)28840 30384 y Fe(k)29481 29810 y Fv(h)30255 28465 y Ff(\264)18890 31652 y Fp(solv)-34 b(e)404 b Fv(M)23183 31840 y Fl(LS)24500 31652 y Fv(u)25274 31849 y Fl(k)26180 31652 y Fp(=)336 b Fv(b)28233 31840 y Fl(LS)30359 31652 y Fp(w.r.t.)538 b Fv(u)34503 31849 y Fl(k)18890 33157 y Fv(a)19568 33354 y Fl(k)20473 33157 y Fj(\303)337 b Fv(a)22700 33339 y Fm(0)23495 33157 y Fp(+)269 b Fv(F)g Fj(\244)h Fv(u)27503 33354 y Fl(k)18890 35183 y Fi(\257)19576 35380 y Fl(k)24 b Fm(+1)21683 35183 y Fj(\303)23232 33706 y Ff(q)p 24444 33706 4293 49 v 1477 x Fv(a)25122 34766 y Fl(T)25122 35575 y(k)26056 34873 y Fp(^)25856 35183 y Fv(\247)26863 35365 y Fo(x)27490 35183 y Fv(a)28168 35380 y Fl(k)18890 37413 y Fi(\264)19492 37610 y Fl(k)g Fm(+1)21600 37413 y Fj(\303)23149 35936 y Ff(q)p 24361 35936 4308 49 v 1477 x Fv(a)25039 36996 y Fl(T)25039 37805 y(k)25972 37104 y Fp(^)25772 37413 y Fv(\247)26779 37595 y Fo(y)27422 37413 y Fv(a)28100 37610 y Fl(k)18890 39124 y Fi(k)375 b Fj(\303)337 b Fi(k)307 b Fp(+)269 b(1)2720 40629 y(Un)-34 b(til)405 b(stop)g(criterion)e (satis\257ed)2720 43640 y(Assign)12689 b Fv(a)336 b Fj(\303)h Fv(a)22131 43837 y Fl(k)24 b Fn(\241)p Fm(1)18890 45255 y Fi(b)337 b Fj(\303)g Fv(a)21974 44815 y Fl(T)22772 45238 y Fp(^)22707 45255 y Fv(x)269 b Fj(\241)25913 44707 y Fl(\257)26441 44863 y Fe(k)p 25057 44976 2749 49 v 25057 45673 a Fl(\257)25585 45829 y Fe(k)26093 45673 y Fm(+)p Fl(\264)27297 45829 y Fe(k)18890 46899 y Fi(\267)337 b Fj(\303)22746 46421 y Fm(1)p 21607 46619 V 21607 47317 a Fl(\257)22135 47473 y Fe(k)22643 47317 y Fm(+)p Fl(\264)23847 47473 y Fe(k)18890 48707 y Fi(\256)342 b Fj(\303)22289 48229 y Fl(\267)22831 47917 y Fc(2)p 21688 48427 2206 49 v 21688 49156 a Fm(1+)p Fl(\267)23432 48904 y Fc(2)p 2056 50798 47889 45 v 0 53079 a Fp(T)-101 b(able)404 b(1:)606 b(Algorithmic)644 b(description)h(of)g(the)g(iterativ)-34 b(e)644 b(pro)34 b(cedure)644 b(to)h(solv)-34 b(e)644 b(the)h(optimization)g(problem)4882 54585 y(for)791 b(a)g(linear)f (MPM.)h(The)h(stop)f(criterion)g(can)g(b)34 b(e)790 b(c)-34 b(hosen)792 b(b)-34 b(y)791 b(the)h(user,)887 b(e.g.,)f(stopping)4882 56090 y(when)606 b(the)g(relativ)-34 b(e)604 b(c)-34 b(hange)606 b(in)f Fi(\257)21560 56287 y Fl(k)22532 56090 y Fp(+)403 b Fi(\264)24480 56287 y Fl(k)25654 56090 y Fp(is)605 b(small)g(enough)h(or)f(a)g(maxim)-34 b(um)606 b(n)-34 b(um)g(b)34 b(er)606 b(of)f(iter-)4882 57596 y(ations)598 b(has)g(b)34 b(een)598 b(reac)-34 b(hed.)1118 b(A)598 b(Matlab)g(implemen)-34 b(tation)598 b(of)g(this)g(algorithm)g (is)f(a)-34 b(v)-67 b(ailable)597 b(on)4882 59101 y Fh(http://r)-62 b(ob)g(otics.e)g(e)g(cs.b)g(erkeley.e)g(du/)-86 b Fp(~)-99 b Fh(gert/)p Fp(.)25697 72000 y(7)p eop %%Page: 8 8 8 7 bop 0 1218 a Fh(is)433 b(uniquely)e(attaine)-62 b(d.)555 b(L)-62 b(et)434 b Fj(f)p Fv(x)14679 1415 y Fl(k)15247 1218 y Fj(g)g Fh(b)-62 b(e)433 b(the)f(se)-62 b(quenc)g(e)431 b(gener)-62 b(ate)g(d)432 b(by)h(the)f(blo)-62 b(ck)432 b(c)-62 b(o)g(or)g(dinate)431 b(desc)-62 b(ent)432 b(metho)-62 b(d)8376 3308 y Fv(x)9112 2799 y Fl(k)24 b Fm(+1)9112 3670 y Fl(i)11219 3308 y Fp(=)337 b(arg)467 b(min)14405 4125 y Fl(\273)14819 4260 y Fe(i)15170 4125 y Fn(2)p Fl(X)16571 4260 y Fe(i)17124 3308 y Fi(f)130 b Fp(\()p Fv(x)19054 2799 y Fl(k)24 b Fm(+1)19054 3655 y(1)20826 3308 y Fi(;)202 b(:)g(:)g(:)f(;)h Fv(x)24256 2799 y Fl(k)24 b Fm(+1)24256 3670 y Fl(i)p Fn(\241)p Fm(1)26027 3308 y Fi(;)202 b(\273)27096 3490 y Fl(i)27471 3308 y Fi(;)g Fv(x)28746 2808 y Fl(k)28746 3608 y(i)p Fm(+1)30324 3308 y Fi(;)g(:)g(:)g(:)f(;)h Fv(x)33754 2808 y Fl(k)33754 3608 y(m)34641 3308 y Fp(\))p Fi(;)1441 b(i)336 b Fp(=)g(1)p Fi(;)202 b(:)g(:)g(:)f(;)h(m:)0 6024 y Fh(Then,)433 b(every)f(limit)h (p)-62 b(oint)433 b(of)g Fj(f)p Fv(x)15452 6221 y Fl(k)16020 6024 y Fj(g)h Fh(is)f(a)g(stationary)e(p)-62 b(oint.)0 7998 y Fp(W)-101 b(e)454 b(can)g(tak)-34 b(e)455 b Fi(X)550 b Fp(to)454 b(b)34 b(e)455 b(an)-34 b(y)454 b Fi(s)p Fp(-sublev)-34 b(el)455 b(set)f(of)h(the)g(original)f(problem)g (\(12\),)468 b(whic)-34 b(h)455 b(will)f(b)34 b(e)454 b(b)34 b(ounded)456 b(and)0 9504 y(closed)404 b(\(as)g(the)h(in)-34 b(tersection)404 b(of)h(a)f(cone)g(with)h(a)f(h)-34 b(yp)34 b(erplane\),)404 b(and)h(tak)-34 b(e)404 b Fi(\257)64 b(;)202 b(\264)380 b Fj(2)337 b Fp([0)p Fi(;)202 b(s)p Fp(].)1882 11009 y(Iterativ)-34 b(e)459 b(pro)34 b(cedures)459 b(to)g(solv)-34 b(e)459 b(w)-34 b(eigh)g(ted)461 b(least-squares)f (problems)f(ha)-34 b(v)g(e)460 b(b)34 b(een)459 b(applied)h(b)34 b(efore)459 b(in)h(the)0 12515 y(SVM)453 b(literature)f(to)g(ac)-34 b(hiev)g(e)452 b(an)h(appro)-34 b(ximate)453 b(solution)g(as)g(in)f (the)h(Least)f(Squares)h(SVM)g(in)f(Suyk)-34 b(ens)453 b(and)0 14020 y(V)-101 b(andew)-34 b(alle)405 b(\(1999\))g(or)f(an)g (exact)g(solution)h(as)f(in)g(P)-34 b(\266)-572 b(erez-Cruz)403 b(et)h(al.)g(\(2001\).)0 17126 y Fv(2.4)697 b(Geometric)464 b(In)-39 b(terpretation)0 19271 y Fp(Problem)371 b(\(6\))i(admits)f(an) g(app)34 b(ealing)372 b(geometric)f(in)-34 b(terpretation,)378 b(whic)-34 b(h)373 b(w)-34 b(e)372 b(obtain)h(via)e(con)-34 b(v)g(ex)371 b(dualit)-34 b(y)-101 b(.)529 b(T)-101 b(o)0 20777 y(obtain)405 b(the)g(dual)f(of)h(the)f(problem,)g(w)-34 b(e)405 b(express)e(\(12\))i(as)f(a)g(constrained)h(minimax)f(problem:) 5883 22867 y(min)6629 23595 y Fo(a)8377 22867 y Fp(max)8374 23684 y Fl(\270;)p Fo(u)p Fl(;)p Fo(v)11108 22867 y Fv(u)11882 22367 y Fl(T)12616 22867 y Fv(\247)13623 23049 y Fo(x)14251 22367 y Fm(1)p Fl(=)p Fm(2)15717 22867 y Fv(a)269 b Fp(+)g Fv(v)18631 22367 y Fl(T)19365 22867 y Fv(\247)20372 23049 y Fo(y)21015 22367 y Fm(1)p Fl(=)p Fm(2)22482 22867 y Fv(a)g Fp(+)g Fi(\270)p Fp(\(1)h Fj(\241)f Fv(a)28585 22367 y Fl(T)29318 22867 y Fp(\()29854 22850 y(\271)29789 22867 y Fv(x)h Fj(\241)32081 22850 y Fp(\271)32007 22867 y Fv(y)19 b Fp(\)\))741 b(:)g Fj(k)p Fv(u)p Fj(k)37509 23049 y Fm(2)38372 22867 y Fj(\267)336 b Fp(1)p Fi(;)471 b Fj(k)p Fv(v)19 b Fj(k)43032 23049 y Fm(2)43895 22867 y Fj(\267)336 b Fp(1)p Fi(:)0 25719 y Fp(The)270 b(ab)34 b(o)-34 b(v)g(e)270 b(is)f(based)h(on)g(the)f(follo)-34 b(wing)271 b(expression)e(for)g(the)h(2-norm)g(of)g(a)f(v)-34 b(ector)269 b Fv(z)p Fp(:)471 b Fj(k)p Fv(z)p Fj(k)40775 25901 y Fm(2)41637 25719 y Fp(=)336 b(max)p Fj(f)p Fv(u)46552 25279 y Fl(T)47287 25719 y Fv(z)605 b Fp(:)h Fj(k)p Fv(u)p Fj(k)51441 25901 y Fm(2)52304 25719 y Fj(\267)0 27224 y Fp(1)p Fj(g)p Fp(.)599 b(Since)424 b(the)g(equalit)-34 b(y-constrained)425 b(original)f(\(primal\))g(problem)g(\(6\))h(is)f (feasible)g(\(as)h(implied)f(b)-34 b(y)48083 27207 y(\271)48018 27224 y Fv(x)370 b Fj(6)p Fp(=)50511 27207 y(\271)50437 27224 y Fv(y)19 b Fp(\),)0 28730 y(standard)311 b(results)e(in)h(con) -34 b(v)g(ex)309 b(optimization)h(\(see)f(e.g.,)618 b(Bo)-34 b(yd)309 b(and)h(V)-101 b(anden)-34 b(b)34 b(erghe,)311 b(2001\))f(imply)e(that)j(w)-34 b(e)310 b(can)0 30235 y(exc)-34 b(hange)422 b(the)g(\\min")h(and)f(\\max")g(in)g(the)h(ab)34 b(o)-34 b(v)g(e)422 b(and)g(still)g(obtain)h(the)f(same)g(optimal)g(v) -67 b(alue.)591 b(Minimizing)0 31741 y(with)405 b(resp)34 b(ect)403 b(to)i Fv(a)f Fp(in)g(the)g(ab)34 b(o)-34 b(v)g(e)405 b(for)f(\257xed)g Fv(u)p Fi(;)202 b Fv(v)19 b Fi(;)202 b(\270)404 b Fp(yields)g(the)g(so-called)g(dual)g(function:)11876 34526 y Fi(g)43 b Fp(\()p Fv(u)p Fi(;)202 b Fv(v)19 b Fi(;)202 b(\270)p Fp(\))338 b(=)18371 32817 y Ff(\275)19833 33817 y Fi(\270)2555 b Fp(if)404 b Fi(\270)24978 33800 y Fp(\271)24913 33817 y Fv(x)269 b Fj(\241)g Fv(\247)28137 33999 y Fo(x)28765 33336 y Fm(1)p Fl(=)p Fm(2)30232 33817 y Fv(u)337 b Fp(=)f Fi(\270)33403 33800 y Fp(\271)33329 33817 y Fv(y)289 b Fp(+)269 b Fv(\247)36573 33999 y Fo(y)37216 33336 y Fm(1)p Fl(=)p Fm(2)38682 33817 y Fv(v)19833 35323 y Fj(\2411)1107 b Fp(otherwise.)0 37312 y(The)500 b(dual)g(problem)f (consists)h(in)f(maximizing)g(the)g(ab)34 b(o)-34 b(v)g(e)500 b(function)g(o)-34 b(v)g(er)500 b(the)f(norm)h(constrain)-34 b(ts)500 b(on)g Fv(u)p Fi(;)202 b Fv(v)19 b Fp(.)0 38818 y(Precisely)-101 b(,)402 b(the)i(dual)h(problem)f(is)9653 40908 y(max)9650 41725 y Fl(\270;)p Fo(u)p Fl(;)p Fo(v)12383 40908 y Fi(\270)741 b Fp(:)f Fj(k)p Fv(u)p Fj(k)16894 41090 y Fm(2)17757 40908 y Fj(\267)337 b Fp(1)p Fi(;)875 b Fj(k)p Fv(v)19 b Fj(k)22822 41090 y Fm(2)23685 40908 y Fj(\267)336 b Fp(1)p Fi(;)875 b(\270)27554 40891 y Fp(\271)27489 40908 y Fv(x)269 b Fj(\241)h Fv(\247)30714 41090 y Fo(x)31341 40408 y Fm(1)p Fl(=)p Fm(2)32808 40908 y Fv(u)337 b Fp(=)f Fi(\270)35979 40891 y Fp(\271)35905 40908 y Fv(y)289 b Fp(+)269 b Fv(\247)39149 41090 y Fo(y)39792 40408 y Fm(1)p Fl(=)p Fm(2)41258 40908 y Fv(v)19 b Fi(:)0 43556 y Fp(A)-34 b(t)405 b(the)f(optim)-34 b(um,)406 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)15590 45647 y Fi(\270)16297 45829 y Fn(\244)17159 45647 y Fp(=)337 b Fj(k)p Fv(\247)20052 45829 y Fo(x)20680 45147 y Fm(1)p Fl(=)p Fm(2)22146 45647 y Fv(a)22824 45829 y Fn(\244)23350 45647 y Fj(k)23956 45829 y Fm(2)24751 45647 y Fp(+)269 b Fj(k)p Fv(\247)27576 45829 y Fo(y)28219 45147 y Fm(1)p Fl(=)p Fm(2)29686 45647 y Fv(a)30364 45829 y Fn(\244)30889 45647 y Fj(k)31495 45829 y Fm(2)32358 45647 y Fp(=)336 b(1)p Fi(=\267)35547 45829 y Fn(\244)36073 45647 y Fi(;)0 47738 y Fp(since)404 b(the)g(optimal)h(v)-67 b(alues)403 b(of)i(the)f(primal)g(and)h(dual)f (problem)h(coincide.)1882 49243 y(De\257ne)f Fi(\267)337 b Fp(:=)f(1)p Fi(=\270)p Fp(;)404 b(c)-34 b(hanging)405 b(the)g(sign)f(of)h Fv(u)p Fp(,)f(w)-34 b(e)404 b(obtain)10390 51334 y(min)10272 52061 y Fl(\267;)p Fo(u)p Fl(;)p Fo(v)12999 51334 y Fi(\267)741 b Fp(:)15580 51317 y(\271)15515 51334 y Fv(x)269 b Fp(+)h Fv(\247)18740 51516 y Fo(x)19367 50833 y Fm(1)p Fl(=)p Fm(2)20834 51334 y Fv(u)337 b Fp(=)23299 51317 y(\271)23224 51334 y Fv(y)289 b Fp(+)269 b Fv(\247)26468 51516 y Fo(y)27111 50833 y Fm(1)p Fl(=)p Fm(2)28577 51334 y Fv(v)19 b Fi(;)876 b Fj(k)p Fv(u)p Fj(k)32531 51516 y Fm(2)33394 51334 y Fj(\267)336 b Fi(\267;)876 b Fj(k)p Fv(v)19 b Fj(k)38551 51516 y Fm(2)39413 51334 y Fj(\267)337 b Fi(\267:)8117 b Fp(\(14\))1882 53960 y(In)493 b(the)g(dual)g(form)g (\(14\),)515 b(our)493 b(problem)g(admits)g(the)g(follo)-34 b(wing)493 b(geometric)f(in)-34 b(terpretation.)805 b(F)-101 b(or)493 b(giv)-34 b(en)0 55465 y Fi(\267)337 b Fj(\270)g Fp(0,)389 b(consider)d(t)-34 b(w)g(o)387 b(ellipsoids)e(cen)-34 b(tered)386 b(around)h(the)g(means)f(of)g(the)h(t)-34 b(w)g(o)387 b(classes,)i(and)d(shap)34 b(e)387 b(determined)0 56970 y(b)-34 b(y)405 b(their)e(co)-34 b(v)-67 b(ariance)403 b(matrices:)4384 59061 y Fj(E)5024 59243 y Fo(x)5651 59061 y Fp(\()p Fi(\267)p Fp(\))338 b(=)f Fj(f)p Fv(x)f Fp(=)11931 59044 y(\271)11867 59061 y Fv(x)269 b Fp(+)g Fv(\247)15091 59243 y Fo(x)15718 58561 y Fm(1)p Fl(=)p Fm(2)17185 59061 y Fv(u)741 b Fp(:)g Fj(k)p Fv(u)p Fj(k)21764 59243 y Fm(2)22627 59061 y Fj(\267)336 b Fi(\267)p Fj(g)p Fi(;)876 b Fj(E)27063 59243 y Fo(y)27705 59061 y Fp(\()p Fi(\267)p Fp(\))338 b(=)f Fj(f)p Fv(y)355 b Fp(=)34014 59044 y(\271)33940 59061 y Fv(y)288 b Fp(+)269 b Fv(\247)37183 59243 y Fo(y)37826 58561 y Fm(1)p Fl(=)p Fm(2)39293 59061 y Fv(v)760 b Fp(:)740 b Fj(k)p Fv(v)19 b Fj(k)43833 59243 y Fm(2)44696 59061 y Fj(\267)336 b Fi(\267)p Fj(g)p Fi(:)2229 b Fp(\(15\))0 61152 y(These)407 b(sets)h(corresp)34 b(ond)407 b(to)g(the)g(p)34 b(oin)-34 b(ts)409 b(whose)e(Mahalanobis)h(distances) g(to)f(the)h(class)e(means)i(are)e(less)h(than)0 62657 y(a)d(sp)34 b(eci\257ed)404 b(n)-34 b(um)g(b)34 b(er)405 b Fi(\267)p Fp(.)1882 64163 y(Clearly)-101 b(,)419 b(for)e Fi(\267)h Fp(large)e(enough,)421 b(these)c(sets)h(in)-34 b(tersect.)577 b(Problem)417 b(\(14\))h(amoun)-34 b(ts)419 b(to)e(\257nding)i(the)e(smallest)0 65668 y Fi(\267)532 b Fp(for)f(whic)-34 b(h)532 b(these)g(ellipsoids)e(in)-34 b(tersect.)920 b(F)-101 b(or)532 b(the)f(optimal)h Fi(\267)p Fp(,)563 b(the)532 b(ellipsoids)f(will)f(b)34 b(e)531 b(tangen)-34 b(t)533 b(to)f(eac)-34 b(h)0 67174 y(other.)569 b(The)415 b(minimax)f(probabilistic)g(decision)g(h)-34 b(yp)34 b(erplane)414 b(is)g(then)h(the)g(common)f(tangen)-34 b(t)416 b(to)f(the)g(optimal)0 68679 y(ellipsoids.)538 b(This)405 b(is)f(illustrated)g(in)g(Figure)g(1.)25697 72000 y(8)p eop %%Page: 9 9 9 8 bop 8677 24194 a gsave currentpoint currentpoint translate 0 neg rotate neg exch neg exch translate 8677 24194 a @beginspecial 63 @llx 223 @lly 547 @urx 561 @ury 3118 @rwi @setspecial %%BeginDocument: geometr_interpret.ps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: C:\research\MPM\figure\geometr_interpret.eps %%CreationDate: 03/15/2002 11:25:22 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 63 223 547 561 %%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 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 63 223 547 561 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0204 7344 csm 557 601 5814 4057 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 6913 5185 PR 6 w 0 3570 5356 0 0 -3570 899 4286 4 MP PP -5356 0 0 3570 5356 0 0 -3570 899 4286 5 MP stroke 4 w DO SO 6 w 0 sg 899 716 mt 6255 716 L 899 4286 mt 6255 4286 L 6255 4286 mt 6255 716 L 899 4286 mt 899 716 L 899 4286 mt 6255 4286 L 899 4286 mt 899 716 L 899 4286 mt 899 4233 L 899 716 mt 899 769 L %%IncludeResource: font Helvetica 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899 1311 mt 952 1311 L 6255 1311 mt 6201 1311 L 798 1355 mt (8) s 899 716 mt 952 716 L 6255 716 mt 6201 716 L 731 760 mt (10) s 899 716 mt 6255 716 L 899 4286 mt 6255 4286 L 6255 4286 mt 6255 716 L 899 4286 mt 899 716 L gs 899 716 5357 3571 MR c np gr /c8 { 0.000000 0.000000 1.000000 sr} bdef c8 2079 1828 mt 2129 1878 L 2129 1828 mt 2079 1878 L 2118 2356 mt 2168 2406 L 2168 2356 mt 2118 2406 L 2183 2075 mt 2233 2125 L 2233 2075 mt 2183 2125 L 2131 1966 mt 2181 2016 L 2181 1966 mt 2131 2016 L 2136 2055 mt 2186 2105 L 2186 2055 mt 2136 2105 L 1828 2583 mt 1878 2633 L 1878 2583 mt 1828 2633 L 1783 2535 mt 1833 2585 L 1833 2535 mt 1783 2585 L 1786 2395 mt 1836 2445 L 1836 2395 mt 1786 2445 L 2060 2461 mt 2110 2511 L 2110 2461 mt 2060 2511 L 2143 2794 mt 2193 2844 L 2193 2794 mt 2143 2844 L 2085 1870 mt 2135 1920 L 2135 1870 mt 2085 1920 L 2196 1937 mt 2246 1987 L 2246 1937 mt 2196 1987 L 1979 2438 mt 2029 2488 L 2029 2438 mt 1979 2488 L 2483 1491 mt 2533 1541 L 2533 1491 mt 2483 1541 L 2467 2115 mt 2517 2165 L 2517 2115 mt 2467 2165 L 1777 1718 mt 1827 1768 L 1827 1718 mt 1777 1768 L 2099 1740 mt 2149 1790 L 2149 1740 mt 2099 1790 L 2230 2279 mt 2280 2329 L 2280 2279 mt 2230 2329 L 2409 2316 mt 2459 2366 L 2459 2316 mt 2409 2366 L 2463 1667 mt 2513 1717 L 2513 1667 mt 2463 1717 L 2328 1887 mt 2378 1937 L 2378 1887 mt 2328 1937 L 1704 2551 mt 1754 2601 L 1754 2551 mt 1704 2601 L 1485 2619 mt 1535 2669 L 1535 2619 mt 1485 2669 L 2012 1749 mt 2062 1799 L 2062 1749 mt 2012 1799 L 2085 2374 mt 2135 2424 L 2135 2374 mt 2085 2424 L 1682 2525 mt 1732 2575 L 1732 2525 mt 1682 2575 L 1934 2220 mt 1984 2270 L 1984 2220 mt 1934 2270 L 1976 1890 mt 2026 1940 L 2026 1890 mt 1976 1940 L 2653 2156 mt 2703 2206 L 2703 2156 mt 2653 2206 L 1830 1876 mt 1880 1926 L 1880 1876 mt 1830 1926 L 1737 2372 mt 1787 2422 L 1787 2372 mt 1737 2422 L 2571 2256 mt 2621 2306 L 2621 2256 mt 2571 2306 L 2605 2240 mt 2655 2290 L 2655 2240 mt 2605 2290 L 2169 2954 mt 2219 3004 L 2219 2954 mt 2169 3004 L 1786 1837 mt 1836 1887 L 1836 1837 mt 1786 1887 L 1942 2244 mt 1992 2294 L 1992 2244 mt 1942 2294 L 1533 2341 mt 1583 2391 L 1583 2341 mt 1533 2391 L 1711 2622 mt 1761 2672 L 1761 2622 mt 1711 2672 L 1787 2249 mt 1837 2299 L 1837 2249 mt 1787 2299 L 2405 1963 mt 2455 2013 L 2455 1963 mt 2405 2013 L 2011 1057 mt 2061 1107 L 2061 1057 mt 2011 1107 L 2235 1618 mt 2285 1668 L 2285 1618 mt 2235 1668 L 2547 1869 mt 2597 1919 L 2597 1869 mt 2547 1919 L 2170 2290 mt 2220 2340 L 2220 2290 mt 2170 2340 L 1773 3114 mt 1823 3164 L 1823 3114 mt 1773 3164 L 2027 1444 mt 2077 1494 L 2077 1444 mt 2027 1494 L 1777 2705 mt 1827 2755 L 1827 2705 mt 1777 2755 L 1918 2012 mt 1968 2062 L 1968 2012 mt 1918 2062 L 2226 1835 mt 2276 1885 L 2276 1835 mt 2226 1885 L 2005 2701 mt 2055 2751 L 2055 2701 mt 2005 2751 L 1602 3087 mt 1652 3137 L 1652 3087 mt 1602 3137 L 2549 1655 mt 2599 1705 L 2599 1655 mt 2549 1705 L 1994 2251 mt 2044 2301 L 2044 2251 mt 1994 2301 L 2228 1733 mt 2278 1783 L 2278 1733 mt 2228 1783 L 2592 2401 mt 2642 2451 L 2642 2401 mt 2592 2451 L 2386 1971 mt 2436 2021 L 2436 1971 mt 2386 2021 L 2511 2193 mt 2561 2243 L 2561 2193 mt 2511 2243 L 1697 2233 mt 1747 2283 L 1747 2233 mt 1697 2283 L 1492 1985 mt 1542 2035 L 1542 1985 mt 1492 2035 L 2187 2103 mt 2237 2153 L 2237 2103 mt 2187 2153 L 1765 2805 mt 1815 2855 L 1815 2805 mt 1765 2855 L 2035 2440 mt 2085 2490 L 2085 2440 mt 2035 2490 L 2212 1404 mt 2262 1454 L 2262 1404 mt 2212 1454 L 2160 2217 mt 2210 2267 L 2210 2217 mt 2160 2267 L 1720 2328 mt 1770 2378 L 1770 2328 mt 1720 2378 L 2339 2057 mt 2389 2107 L 2389 2057 mt 2339 2107 L 2251 1392 mt 2301 1442 L 2301 1392 mt 2251 1442 L 1957 3119 mt 2007 3169 L 2007 3119 mt 1957 3169 L 1781 2508 mt 1831 2558 L 1831 2508 mt 1781 2558 L 2273 2848 mt 2323 2898 L 2323 2848 mt 2273 2898 L 2064 2155 mt 2114 2205 L 2114 2155 mt 2064 2205 L 2016 2377 mt 2066 2427 L 2066 2377 mt 2016 2427 L 2193 1784 mt 2243 1834 L 2243 1784 mt 2193 1834 L 1979 2390 mt 2029 2440 L 2029 2390 mt 1979 2440 L 2110 2052 mt 2160 2102 L 2160 2052 mt 2110 2102 L 2025 2307 mt 2075 2357 L 2075 2307 mt 2025 2357 L 2013 2446 mt 2063 2496 L 2063 2446 mt 2013 2496 L 2445 1929 mt 2495 1979 L 2495 1929 mt 2445 1979 L 2295 1215 mt 2345 1265 L 2345 1215 mt 2295 1265 L 1597 2105 mt 1647 2155 L 1647 2105 mt 1597 2155 L 2320 3114 mt 2370 3164 L 2370 3114 mt 2320 3164 L 1961 2746 mt 2011 2796 L 2011 2746 mt 1961 2796 L 2369 2759 mt 2419 2809 L 2419 2759 mt 2369 2809 L 1979 1534 mt 2029 1584 L 2029 1534 mt 1979 1584 L 1354 2262 mt 1404 2312 L 1404 2262 mt 1354 2312 L 2229 1830 mt 2279 1880 L 2279 1830 mt 2229 1880 L 1874 2601 mt 1924 2651 L 1924 2601 mt 1874 2651 L 1840 2257 mt 1890 2307 L 1890 2257 mt 1840 2307 L 1822 2409 mt 1872 2459 L 1872 2409 mt 1822 2459 L 1688 2712 mt 1738 2762 L 1738 2712 mt 1688 2762 L 2150 2806 mt 2200 2856 L 2200 2806 mt 2150 2856 L 2254 2366 mt 2304 2416 L 2304 2366 mt 2254 2416 L 1755 2138 mt 1805 2188 L 1805 2138 mt 1755 2188 L 1836 1908 mt 1886 1958 L 1886 1908 mt 1836 1958 L 1959 2444 mt 2009 2494 L 2009 2444 mt 1959 2494 L 1708 2974 mt 1758 3024 L 1758 2974 mt 1708 3024 L 1542 2510 mt 1592 2560 L 1592 2510 mt 1542 2560 L 2347 1642 mt 2397 1692 L 2397 1642 mt 2347 1692 L 2318 1745 mt 2368 1795 L 2368 1745 mt 2318 1795 L 2001 2657 mt 2051 2707 L 2051 2657 mt 2001 2707 L gs 899 716 5357 3571 MR c np gr 24 W 4276 2433 PD 24 W 3943 2470 PD 24 W 4839 3014 PD 24 W 4404 2911 PD 24 W 4492 2721 PD 24 W 4344 2776 PD 24 W 4963 3075 PD 24 W 4380 3069 PD 24 W 4538 3037 PD 24 W 4117 3067 PD 24 W 4312 2662 PD 24 W 4510 3149 PD 24 W 3949 2975 PD 24 W 5162 2567 PD 24 W 3227 1980 PD 24 W 3623 2035 PD 24 W 4946 3067 PD 24 W 4155 3078 PD 24 W 4791 2544 PD 24 W 4512 3168 PD 24 W 3814 2600 PD 24 W 3544 2308 PD 24 W 4965 2797 PD 24 W 4930 3230 PD 24 W 4725 2924 PD 24 W 4433 2517 PD 24 W 3750 2554 PD 24 W 5054 2898 PD 24 W 4827 2972 PD 24 W 3878 2614 PD 24 W 3976 2328 PD 24 W 4282 2746 PD 24 W 4006 2309 PD 24 W 3563 2427 PD 24 W 3209 2383 PD 24 W 4332 2894 PD 24 W 3672 2828 PD 24 W 4359 2834 PD 24 W 5527 3040 PD 24 W 4096 2477 PD 24 W 4843 2527 PD 24 W 4771 2938 PD 24 W 4342 3130 PD 24 W 5064 3193 PD 24 W 4425 2296 PD 24 W 4545 2989 PD 24 W 5282 3359 PD 24 W 4789 2921 PD 24 W 5964 3124 PD 24 W 4298 2781 PD 24 W 4611 3414 PD 24 W 4986 3179 PD 24 W 3981 2731 PD 24 W 4976 3430 PD 24 W 4750 2476 PD 24 W 4432 3135 PD 24 W 5147 3111 PD 24 W 5542 3219 PD 24 W 3692 2378 PD 24 W 4656 2702 PD 24 W 4757 2890 PD 24 W 4569 2939 PD 24 W 3973 2670 PD 24 W 5173 2628 PD 24 W 5004 2831 PD 24 W 4964 3045 PD 24 W 4974 2631 PD 24 W 5337 3102 PD 24 W 3773 2764 PD 24 W 4136 2723 PD 24 W 3828 1998 PD 24 W 3783 1988 PD 24 W 3920 2644 PD 24 W 4188 2501 PD 24 W 4138 2717 PD 24 W 5128 2943 PD 24 W 4559 2893 PD 24 W 4980 3261 PD 24 W 5740 3520 PD 24 W 4075 2987 PD 24 W 4639 2688 PD 24 W 4010 2297 PD 24 W 4838 2740 PD 24 W 3888 3102 PD 24 W 4778 2943 PD 24 W 4205 3135 PD 24 W 4172 2379 PD 24 W 4728 2931 PD 24 W 5399 3299 PD 24 W 4296 2575 PD 24 W 4679 2820 PD 24 W 4667 2567 PD 24 W 3575 2437 PD 24 W 3297 2715 PD 24 W 4092 2850 PD 24 W 4931 3179 PD 24 W 4257 2513 PD 24 W 5135 2601 PD 24 W 4194 2992 PD 24 W 4304 2778 PD gs 899 716 5357 3571 MR c np gr /c9 { 1.000000 0.000000 0.000000 sr} bdef c9 36 36 2073 2236 FO gs 899 716 5357 3571 MR c np gr 36 36 4466 2797 FO gs 899 716 5357 3571 MR c np 4 -27 5 -28 7 -27 7 -27 9 -27 9 -27 10 -26 12 -26 12 -25 13 -24 14 -24 14 -22 15 -22 16 -21 16 -19 17 -19 17 -16 17 -16 18 -14 18 -13 18 -11 18 -9 18 -8 18 -6 17 -5 18 -3 17 -1 17 1 16 2 16 4 16 6 14 7 15 9 13 11 12 12 12 13 11 15 10 17 9 18 8 19 6 20 6 21 5 23 4 23 2 24 1 25 1 26 -1 26 -2 26 -3 27 -5 27 -5 28 -6 27 -8 27 -8 27 -10 27 -10 26 -11 26 -13 25 -13 24 -13 24 -15 23 -15 21 -15 21 -17 19 -16 19 -17 17 -18 15 -17 14 -18 13 -18 11 -18 9 -18 8 -18 6 -18 5 -18 3 -17 1 -17 -1 -16 -2 -16 -4 -15 -6 -15 -7 -14 -9 -13 -10 -13 -13 -12 -13 -10 -15 -10 -17 -9 -18 -8 -19 -7 -20 -6 -21 -5 -23 -3 -23 -3 -24 -1 -25 0 -26 1 -26 2 -26 2357 2150 100 MP stroke 3 -27 2354 2177 2 MP stroke -8 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29858 68679 a Fv(a)30536 68329 y Fl(T)31269 68679 y Fv(\247)32276 68861 y Fo(x)32904 68679 y Fv(a)o Fi(:)25394 72000 y Fp(10)p eop %%Page: 11 11 11 10 bop 0 1218 a Fp(The)469 b(same)f(can)g(b)34 b(e)468 b(done)g(for)g(the)h(second)f(constrain)-34 b(t)470 b(in)e(\(1\).)731 b(This)469 b(leads)f(to)g(the)h(follo)-34 b(wing)468 b(optimization)0 2723 y(problem:)14511 5118 y(max)14009 5935 y Fl(\256;)p Fo(a)p Fn(6)p Fm(=0)p Fl(;b)17875 5118 y Fi(\256)1112 b Fp(s.t.)1106 b Fj(\241)p Fi(b)270 b Fp(+)f Fv(a)26114 4617 y Fl(T)26912 5101 y Fp(\271)26847 5118 y Fv(x)337 b Fj(\270)f Fp(\251)30074 4617 y Fn(\241)p Fm(1)31332 5118 y Fp(\()p Fi(\256)t Fp(\))33053 3939 y Ff(p)p 34267 3939 3724 49 v 34267 5118 a Fv(a)34945 4768 y Fl(T)35678 5118 y Fv(\247)36685 5300 y Fo(x)37313 5118 y Fv(a)22491 8225 y Fi(b)270 b Fj(\241)f Fv(a)25171 7724 y Fl(T)25979 8208 y Fp(\271)25904 8225 y Fv(y)356 b Fj(\270)337 b Fp(\251)29151 7724 y Fn(\241)p Fm(1)30409 8225 y Fp(\()p Fi(\256)t Fp(\))32130 6768 y Ff(q)p 33344 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65155 y Fm(1)p 14684 65353 2206 49 v 14684 66083 a(1+)p Fl(\267)16428 65830 y Fc(2)17022 65633 y Fp(,)404 b(if)g Fi(b)g Fp(is)g(feasible)g(for)g(problem)h(\(8\),)f (that)h(is:)14675 68611 y Fv(a)15353 68111 y Fl(T)16161 68594 y Fp(\271)16087 68611 y Fv(y)288 b Fp(+)269 b Fi(\267)19021 67155 y Ff(q)p 20234 67155 3739 49 v 20234 68611 a Fv(a)20912 68261 y Fl(T)21645 68611 y Fv(\247)22652 68793 y Fo(y)23295 68611 y Fv(a)336 b Fj(\267)h Fi(b)g Fj(\267)f Fv(a)28403 68111 y Fl(T)29201 68594 y Fp(\271)29137 68611 y Fv(x)269 b Fj(\241)g Fi(\267)32052 67432 y Ff(p)p 33264 67432 3724 49 v 1179 x Fv(a)33942 68261 y Fl(T)34676 68611 y Fv(\247)35683 68793 y Fo(x)36310 68611 y Fv(a)p Fi(:)25394 72000 y Fp(11)p eop %%Page: 12 12 12 11 bop 0 1218 a Fp(T)-101 b(o)517 b(obtain)g(the)g(smallest)f(p)34 b(ossible)516 b(upp)34 b(er)517 b(b)34 b(ound)517 b(w)-34 b(e)517 b(let)f Fi(\267)h Fp(increase)e(to)i Fi(\267)36824 1406 y Fm(MPM)39243 1218 y Fp(\()p Fv(a)p Fp(\))g(for)f(a)g(giv)-34 b(en)517 b Fv(a)p 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b(p)34 b(oin)-34 b(ts)405 b(using)g(a)f(decision)g(h)-34 b(yp)34 b(erplane)404 b Fj(H)12 b Fp(\()p Fv(a)39158 18165 y Fm(FD)-26 b(A)39158 18904 y Fn(\244)41223 18604 y Fi(;)202 b(b)42282 18165 y Fm(FD)-26 b(A)42282 18904 y Fn(\244)44348 18604 y Fp(\))405 b(where)11533 22190 y Fi(b)12053 21690 y Fm(FD)-26 b(A)12053 22490 y Fn(\244)14456 22190 y Fp(=)337 b(\()p Fv(a)16885 21690 y Fm(FD)-26 b(A)16885 22490 y Fn(\244)18951 22190 y Fp(\))19422 21690 y Fl(T)20221 22173 y Fp(\271)20156 22190 y Fv(x)269 b Fj(\241)g Fi(\267)23071 22378 y Fm(MPM)25490 22190 y Fp(\()p Fv(a)26639 21690 y Fm(FD)-26 b(A)26639 22490 y Fn(\244)28705 22190 y Fp(\))29176 20708 y Ff(q)p 30389 20708 9742 49 v 30389 22190 a Fp(\()p Fv(a)31538 21840 y Fm(FD)g(A)31538 22490 y Fn(\244)33604 22190 y Fp(\))34075 21840 y Fl(T)34809 22190 y Fv(\247)35816 22372 y Fo(x)36443 22190 y Fp(\()p Fv(a)37592 21840 y Fm(FD)g(A)37592 22490 y Fn(\244)39658 22190 y Fp(\))q Fi(:)0 25288 y Fp(This)361 b(not)h(only)e(giv)-34 b(es)361 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b(criterion)e(function)j Fi(\267)49934 33129 y Fm(FD)-26 b(A)0 34446 y Fp(is)535 b(built)h(up)34 b(on)536 b(the)g(in)-34 b(tuition)537 b(that)g(\\separation")f(\(as)g(c)-34 b(haracterized)534 b(b)-34 b(y)536 b(\257rst)g(and)g(second)g(momen)-34 b(ts\))537 b(is)0 35951 y(useful,)387 b(and)c(up)34 b(on)382 b(the)h(desideratum)f(of)h(computational)g(e\261ciency)-101 b(.)530 b(Indeed,)386 b(the)d(solution)g(of)f(\(16\))h(is)e(found)0 37457 y(b)-34 b(y)426 b(solving)f(a)g(generalized)f(eigen)-34 b(v)-67 b(alue)425 b(problem.)602 b(The)426 b(criterion)f(function)h (for)g(MPM,)f(on)h(the)f(other)h(hand,)0 38962 y(di\256ers)362 b(from)h(the)h(FD)-34 b(A)363 b(criterion)f(function)i(in)f(detail,)370 b(but)364 b(due)f(to)h(its)e(similar)g(form)h(preserv)-34 b(es)362 b(the)i(in)-34 b(tuition)0 40468 y(of)569 b(\257nding)h(a)f (direction)g(that)h(separates)f(the)h(data.)1033 b(It)569 b(do)34 b(es)569 b(so)g(while)g(aiming)g(directly)f(at)h(solving)g(the) 0 41973 y(classi\257cation)404 b(problem,)g(and)h(yields)e(an)i (algorithm)f(with)h(similar)e(complexit)-34 b(y)-101 b(.)0 45568 y Fr(3.)648 b(Robustness)500 b(to)e(Estimation)j(Errors)0 47999 y Fp(In)350 b(practical)f(exp)34 b(erimen)-34 b(ts,)360 b(it)349 b(ma)-34 b(y)350 b(w)-34 b(ell)349 b(happ)34 b(en)351 b(that)g(the)f(error)e(rate)i(computed)g(on)g(the)g(test)g (set)g(is)g(greater)0 49504 y(than)583 b(1)388 b Fj(\241)g Fi(\256)6106 49686 y Fn(\244)6632 49504 y Fp(.)1071 b(This)583 b(seems)e(to)h(con)-34 b(tradict)583 b(the)f(previous)g(claim)f(that)i (1)388 b Fj(\241)g Fi(\256)38791 49686 y Fn(\244)39899 49504 y Fp(is)581 b(an)h(upp)34 b(er)582 b(b)34 b(ound)584 b(on)0 51010 y(misclassi\257cation)404 b(error.)1882 52640 y(This)429 b(apparen)-34 b(t)430 b(parado)-34 b(x)430 b(has)f(to)h(do)f(with)g(estimation)h(errors.)611 b(Since)429 b(w)-34 b(e)429 b(do)h(not)f(kno)-34 b(w)430 b(the)f(mean)g(and)0 54145 y(co)-34 b(v)-67 b(ariance)437 b Fh(a)464 b(priori)p Fp(,)445 b(they)438 b(need)g(to)h(b)34 b(e)437 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{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 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 63 223 547 561 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0204 7344 csm 557 601 5814 4057 MR c np 87 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 6913 5185 PR 6 w 0 3570 5356 0 0 -3570 899 4286 4 MP PP -5356 0 0 3570 5356 0 0 -3570 899 4286 5 MP stroke 4 w DO SO 6 w 0 sg 899 716 mt 6255 716 L 899 4286 mt 6255 4286 L 6255 4286 mt 6255 716 L 899 4286 mt 899 716 L 899 4286 mt 6255 4286 L 899 4286 mt 899 716 L 899 4286 mt 899 4233 L 899 716 mt 899 769 L %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 796 4432 mt (-2) s 1494 4286 mt 1494 4233 L 1494 716 mt 1494 769 L 1461 4432 mt (0) s 2089 4286 mt 2089 4233 L 2089 716 mt 2089 769 L 2056 4432 mt (2) s 2684 4286 mt 2684 4233 L 2684 716 mt 2684 769 L 2651 4432 mt (4) s 3279 4286 mt 3279 4233 L 3279 716 mt 3279 769 L 3246 4432 mt (6) s 3874 4286 mt 3874 4233 L 3874 716 mt 3874 769 L 3841 4432 mt (8) s 4469 4286 mt 4469 4233 L 4469 716 mt 4469 769 L 4403 4432 mt (10) s 5064 4286 mt 5064 4233 L 5064 716 mt 5064 769 L 4998 4432 mt (12) s 5659 4286 mt 5659 4233 L 5659 716 mt 5659 769 L 5593 4432 mt (14) s 6255 4286 mt 6255 4233 L 6255 716 mt 6255 769 L 6189 4432 mt (16) s 899 4286 mt 952 4286 L 6255 4286 mt 6201 4286 L 728 4330 mt (-2) s 899 3691 mt 952 3691 L 6255 3691 mt 6201 3691 L 798 3735 mt (0) s 899 3096 mt 952 3096 L 6255 3096 mt 6201 3096 L 798 3140 mt (2) s 899 2501 mt 952 2501 L 6255 2501 mt 6201 2501 L 798 2545 mt (4) s 899 1906 mt 952 1906 L 6255 1906 mt 6201 1906 L 798 1950 mt (6) s 899 1311 mt 952 1311 L 6255 1311 mt 6201 1311 L 798 1355 mt (8) s 899 716 mt 952 716 L 6255 716 mt 6201 716 L 731 760 mt (10) s 899 716 mt 6255 716 L 899 4286 mt 6255 4286 L 6255 4286 mt 6255 716 L 899 4286 mt 899 716 L gs 899 716 5357 3571 MR c np gr /c8 { 1.000000 0.000000 0.000000 sr} bdef c8 36 36 2073 2236 FO gs 899 716 5357 3571 MR c np gr 36 36 4466 2797 FO gs 899 716 5357 3571 MR c np /c9 { 0.000000 1.000000 0.000000 sr} bdef c9 657 -3571 2474 4287 2 MP stroke /c10 { 0.000000 0.000000 1.000000 sr} bdef c10 12 -78 16 -78 18 -79 22 -78 24 -77 27 -77 30 -75 32 -74 35 -72 37 -70 40 -68 41 -65 44 -62 45 -59 46 -56 48 -53 49 -48 50 -45 51 -40 51 -37 51 -31 52 -28 52 -22 51 -18 51 -13 51 -8 49 -3 48 1 47 7 46 11 44 17 43 21 40 25 38 31 36 35 34 39 31 43 28 47 25 52 23 54 20 58 17 62 13 64 11 67 7 69 4 72 0 73 -2 75 -6 76 -9 77 -12 78 -15 78 -19 79 -21 78 -24 77 -28 77 -29 75 -33 74 -35 72 -37 70 -39 68 -42 65 -43 62 -45 59 -47 56 -48 53 -49 48 -49 45 -51 40 -51 37 -52 31 -52 28 -51 22 -52 18 -51 13 -50 8 -50 3 -48 -1 -47 -7 -46 -11 -44 -17 -42 -21 -40 -25 -39 -31 -36 -35 -33 -39 -31 -43 -29 -47 -25 -51 -23 -55 -19 -58 -17 -62 -14 -64 -10 -67 -7 -69 -4 -72 -1 -73 3 -75 5 -76 2889 1990 100 MP stroke 9 -77 2880 2067 2 MP stroke -21 -53 -15 -52 -8 -50 -2 -47 4 -46 10 -43 17 -40 23 -37 29 -34 35 -32 41 -28 47 -24 52 -22 58 -17 62 -15 68 -10 72 -7 76 -3 80 1 84 4 88 8 90 12 93 15 95 19 97 22 99 26 100 29 100 33 101 35 101 38 100 41 100 44 98 46 96 48 95 51 92 52 89 54 87 55 83 57 79 57 75 59 70 59 66 59 61 59 56 59 51 59 45 58 39 57 33 57 27 55 21 53 15 52 8 49 3 48 -5 45 -10 43 -17 40 -23 38 -29 34 -35 31 -41 28 -47 25 -52 21 -57 18 -63 14 -68 11 -72 6 -76 4 -80 -1 -84 -4 -88 -8 -90 -12 -93 -15 -95 -19 -97 -23 -99 -26 -100 -29 -100 -32 -101 -35 -101 -39 -100 -41 -99 -43 -99 -46 -96 -49 -95 -50 -92 -52 -89 -54 -87 -56 -83 -56 -79 -58 -75 -58 -70 -59 -66 -59 -61 -60 -56 -59 -51 -59 -44 -58 -40 -57 -33 -56 5998 3126 100 MP stroke -27 -55 6025 3181 2 MP stroke gr c10 0 sg 3519 4575 mt (z) s %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 96 FMSR 3579 4635 mt (1) s %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 120 FMSR 673 2559 mt -90 rotate (z) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /WindowsLatin1Encoding 96 FMSR 733 2499 mt -90 rotate (2) s 90 rotate gs 899 716 5357 3571 MR c np DA 8 -51 10 -51 12 -51 14 -51 16 -50 18 -50 19 -49 21 -48 23 -47 24 -46 26 -44 27 -42 28 -41 29 -39 31 -36 31 -34 32 -32 32 -29 33 -26 34 -24 33 -20 34 -18 34 -15 33 -12 34 -8 32 -5 33 -2 31 1 31 4 30 7 28 11 28 14 26 17 25 19 23 23 22 25 20 29 19 31 17 33 14 36 13 37 11 40 9 42 7 44 4 45 3 47 0 47 -1 49 -4 50 -6 50 -8 51 -10 51 -12 51 -14 51 -16 50 -17 50 -20 49 -21 48 -23 47 -24 46 -25 44 -27 42 -29 41 -29 39 -30 36 -31 34 -32 32 -33 29 -33 26 -33 24 -34 21 -34 17 -33 15 -34 12 -33 8 -33 5 -32 3 -32 -2 -30 -4 -30 -7 -29 -11 -27 -14 -27 -16 -25 -20 -23 -23 -22 -25 -20 -29 -18 -30 -17 -34 -15 -36 -13 -37 -11 -40 -8 -42 -7 -44 -5 -45 -3 -47 0 -47 2 -49 3 -50 2605 2076 100 MP stroke 6 -50 2599 2126 2 MP stroke /c11 { 1.000000 1.000000 0.000000 sr} bdef c11 10 -51 12 -51 14 -51 16 -50 18 -50 19 -49 21 -48 23 -47 24 -46 26 -44 27 -42 28 -41 29 -39 31 -36 31 -34 32 -32 32 -29 33 -26 34 -24 33 -20 34 -18 34 -15 33 -12 34 -8 32 -5 33 -2 31 1 31 4 30 7 28 11 28 14 26 17 25 19 23 23 22 25 20 29 19 31 17 33 14 36 13 37 11 40 9 42 7 44 4 45 3 47 0 47 -1 49 -4 50 -6 50 -8 51 -10 51 -12 51 -14 51 -16 50 -17 50 -20 49 -21 48 -23 47 -24 46 -25 44 -27 42 -29 41 -29 39 -30 36 -31 34 -32 32 -33 29 -33 26 -33 24 -34 21 -34 17 -33 15 -34 12 -33 8 -33 5 -32 3 -32 -2 -30 -4 -30 -7 -29 -11 -27 -14 -27 -16 -25 -20 -23 -23 -22 -25 -20 -29 -18 -30 -17 -34 -15 -36 -13 -37 -11 -40 -8 -42 -7 -44 -5 -45 -3 -47 0 -47 2 -49 3 -50 6 -50 2599 2126 100 MP 8 -51 1 AP PP SO 0 sg 0 0 8 -51 10 -51 12 -51 14 -51 16 -50 18 -50 19 -49 21 -48 23 -47 24 -46 26 -44 27 -42 28 -41 29 -39 31 -36 31 -34 32 -32 32 -29 33 -26 34 -24 33 -20 34 -18 34 -15 33 -12 34 -8 32 -5 33 -2 31 1 31 4 30 7 28 11 28 14 26 17 25 19 23 23 22 25 20 29 19 31 17 33 14 36 13 37 11 40 9 42 7 44 4 45 3 47 0 47 -1 49 -4 50 -6 50 -8 51 -10 51 -12 51 -14 51 -16 50 -17 50 -20 49 -21 48 -23 47 -24 46 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5464 3012 100 MP stroke -18 -35 5482 3047 2 MP stroke c11 -10 -34 -6 -32 -1 -31 3 -30 6 -28 11 -26 15 -24 19 -22 23 -21 27 -18 30 -16 34 -14 38 -12 41 -9 44 -7 47 -4 49 -2 53 0 54 3 57 5 59 8 61 10 62 12 63 15 64 17 65 19 66 21 65 23 66 25 65 26 65 29 64 30 63 31 62 33 60 34 58 35 56 36 54 37 52 38 49 38 46 38 43 39 39 38 37 39 33 38 29 38 25 37 22 37 18 36 14 35 9 33 6 33 1 31 -3 29 -6 28 -11 26 -15 25 -19 22 -23 20 -27 19 -30 16 -34 14 -38 11 -41 9 -44 7 -46 5 -50 2 -53 -1 -54 -3 -57 -5 -59 -7 -61 -10 -62 -13 -63 -14 -64 -17 -65 -19 -66 -21 -65 -23 -66 -25 -65 -27 -65 -28 -64 -30 -63 -32 -62 -33 -60 -34 -58 -35 -56 -36 -54 -37 -52 -37 -49 -38 -46 -39 -43 -38 -39 -39 -37 -38 -33 -39 -29 -38 -25 -37 -22 -37 -18 -35 5482 3047 100 MP -13 -35 1 AP PP SO 0 sg 0 0 -13 -35 -10 -34 -6 -32 -1 -31 3 -30 6 -28 11 -26 15 -24 19 -22 23 -21 27 -18 30 -16 34 -14 38 -12 41 -9 44 -7 47 -4 49 -2 53 0 54 3 57 5 59 8 61 10 62 12 63 15 64 17 65 19 66 21 65 23 66 25 65 26 65 29 64 30 63 31 62 33 60 34 58 35 56 36 54 37 52 38 49 38 46 38 43 39 39 38 37 39 33 38 29 38 25 37 22 37 18 36 14 35 9 33 6 33 1 31 -3 29 -6 28 -11 26 -15 25 -19 22 -23 20 -27 19 -30 16 -34 14 -38 11 -41 9 -44 7 -46 5 -50 2 -53 -1 -54 -3 -57 -5 -59 -7 -61 -10 -62 -13 -63 -14 -64 -17 -65 -19 -66 -21 -65 -23 -66 -25 -65 -27 -65 -28 -64 -30 -63 -32 -62 -33 -60 -34 -58 -35 -56 -36 -54 -37 -52 -37 -49 -38 -46 -39 -43 -38 -39 -39 -37 -38 -33 -39 -29 -38 -25 -37 5442 2975 100 MP stroke -22 -37 -18 -35 5482 3047 3 MP stroke gr c10 36 36 2073 2236 FO gs 899 716 5357 3571 MR c np gr 36 36 4466 2797 FO gs 899 716 5357 3571 MR c np gr c8 36 36 2587 1752 FO 36 36 2396 1462 FO 36 36 2083 1467 FO 36 36 1765 1766 FO 36 36 1566 2244 FO 36 36 1560 2720 FO 36 36 1750 3010 FO 36 36 2064 3005 FO 36 36 2381 2706 FO 36 36 2581 2228 FO gs 899 716 5357 3571 MR c np DA 5 -27 5 -27 6 -27 8 -27 8 -27 10 -27 10 -26 11 -26 13 -25 13 -25 13 -23 15 -23 15 -22 16 -20 16 -20 16 -18 17 -17 18 -15 18 -15 17 -12 18 -11 18 -10 18 -8 18 -6 18 -4 18 -3 17 -1 17 0 16 3 16 4 15 5 15 8 14 9 14 10 12 12 12 14 11 15 9 16 9 18 8 19 7 21 6 21 5 22 3 24 3 24 1 25 0 25 -1 26 -2 27 -3 27 -4 27 -5 27 -7 28 -7 27 -9 27 -9 26 -10 27 -12 25 -12 25 -13 25 -14 23 -14 23 -15 22 -16 20 -16 20 -17 18 -17 17 -17 16 -18 14 -18 12 -18 11 -18 10 -18 8 -18 6 -17 5 -18 2 -17 2 -17 -1 -16 -2 -16 -4 -16 -6 -14 -7 -14 -9 -14 -11 -12 -12 -12 -14 -11 -15 -10 -16 -9 -18 -7 -19 -7 -20 -6 -22 -5 -22 -3 -23 -3 -25 -1 -25 -1 -25 1 -26 2 -27 2871 1667 100 MP stroke 3 -27 2868 1694 2 MP stroke 4 -27 5 -27 7 -28 7 -27 9 -27 9 -26 10 -27 12 -25 12 -25 13 -25 14 -24 14 -22 15 -22 16 -21 16 -19 17 -18 17 -17 17 -16 18 -14 18 -12 18 -12 18 -9 18 -8 18 -6 17 -5 18 -2 17 -2 17 1 16 2 16 4 16 6 14 7 14 9 14 11 12 12 12 14 11 15 10 16 9 18 7 19 7 20 6 22 5 22 3 23 3 25 1 25 1 25 -1 26 -2 27 -3 27 -5 27 -5 27 -6 27 -8 27 -8 27 -10 27 -10 26 -11 26 -13 25 -13 25 -13 23 -15 23 -15 22 -16 20 -16 20 -16 18 -17 17 -18 15 -18 14 -17 13 -18 11 -18 10 -18 8 -18 6 -18 4 -18 3 -17 1 -17 0 -16 -3 -16 -4 -15 -5 -15 -8 -14 -9 -14 -10 -12 -12 -12 -14 -11 -15 -9 -17 -9 -17 -8 -19 -7 -21 -6 -21 -5 -22 -3 -24 -3 -24 -1 -25 0 -25 1 -27 2 -26 2680 1376 100 MP stroke 3 -27 2677 1403 2 MP stroke 5 -27 5 -27 6 -28 8 -27 8 -27 10 -26 10 -27 11 -25 13 -26 12 -24 14 -24 15 -22 15 -22 15 -21 17 -19 16 -18 17 -17 18 -16 17 -14 18 -13 18 -11 18 -9 18 -8 18 -6 18 -5 18 -3 17 -1 17 1 16 2 16 4 15 6 15 7 14 9 13 11 13 12 12 14 10 15 10 16 9 18 8 19 7 20 6 22 5 22 3 23 3 25 1 24 0 26 -1 26 -2 27 -3 27 -4 27 -5 27 -7 27 -7 27 -9 27 -9 27 -10 26 -12 26 -12 25 -13 24 -14 24 -14 23 -15 22 -16 20 -16 20 -17 18 -17 17 -17 15 -18 14 -18 13 -18 11 -18 10 -18 7 -18 7 -17 4 -18 3 -17 1 -17 0 -16 -3 -16 -4 -16 -5 -14 -8 -15 -9 -13 -10 -12 -12 -12 -14 -11 -15 -10 -17 -9 -17 -8 -19 -6 -21 -6 -21 -5 -23 -4 -23 -2 -24 -1 -25 -1 -25 1 -27 2 -26 2367 1381 100 MP stroke 3 -27 2364 1408 2 MP stroke 4 -27 5 -27 7 -27 7 -28 8 -27 10 -26 10 -27 12 -25 12 -25 13 -25 13 -23 15 -23 15 -22 16 -20 16 -20 17 -18 17 -17 17 -16 18 -14 18 -12 18 -11 18 -10 18 -8 18 -6 17 -5 18 -2 17 -2 17 1 16 2 16 4 16 6 14 7 14 9 14 11 12 12 12 14 11 15 10 16 8 18 8 19 7 20 6 22 5 22 3 24 3 24 1 25 1 25 -1 26 -2 27 -3 27 -5 27 -5 27 -6 27 -8 27 -8 27 -10 27 -10 26 -12 26 -12 25 -13 25 -13 23 -15 23 -15 22 -16 20 -16 20 -16 18 -18 17 -17 15 -18 15 -17 12 -18 11 -19 10 -18 8 -17 6 -18 4 -18 3 -17 1 -17 0 -16 -3 -16 -4 -16 -5 -14 -8 -14 -9 -14 -10 -12 -12 -12 -14 -11 -15 -9 -16 -9 -18 -8 -19 -7 -21 -6 -21 -5 -22 -3 -24 -3 -24 -1 -25 0 -51 2 -27 4 -27 2046 1707 100 MP stroke 4 -28 6 -27 6 -27 8 -27 8 -27 10 -27 10 -26 11 -26 12 -25 13 -24 14 -24 15 -23 15 -21 15 -21 17 -19 16 -19 17 -17 18 -15 17 -14 18 -13 18 -11 18 -10 18 -7 18 -7 18 -4 17 -3 18 -1 17 0 16 3 16 4 15 5 15 8 14 9 13 10 13 12 12 14 10 15 10 17 9 18 8 19 7 20 6 21 4 23 4 23 3 24 1 25 0 26 -1 26 -2 26 -3 27 -4 27 -5 27 -7 28 -7 27 -9 27 -9 27 -11 26 -11 26 -12 25 -13 24 -14 24 -14 22 -15 22 -16 21 -16 19 -17 18 -17 17 -17 16 -18 14 -18 13 -18 11 -18 9 -18 8 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b Fp(\(27\))25394 72000 y(16)p eop %%Page: 17 17 17 16 bop 0 1289 a Fh(if)433 b(the)g(choic)-62 b(e)431 b(of)i Fi(\256)438 b Fh(is)433 b(such)f(that)g Fi(\267)p Fp(\()p Fi(\256)t Fp(\))339 b Fj(\267)d Fi(\263)524 b Fh(or)433 b Fi(\256)342 b Fj(\267)25805 741 y Fl(\263)26280 428 y Fc(2)p 25204 1010 2138 49 v 25204 1738 a Fm(1+)p Fl(\263)26881 1486 y Fc(2)27475 1289 y Fh(.)1882 2930 y(If)403 b(the)f(choic)-62 b(e)401 b(of)h Fi(\256)408 b Fh(is)402 b(not)h(fe)-62 b(asible)400 b(or)j(if)21613 2913 y Fp(\271)21549 2930 y Fv(x)336 b Fp(=)h Fv(0)403 b Fh(\(in)g(this)e(c)-62 b(ase,)407 b(no)d Fi(\256)341 b Fj(2)c Fp(\(0)p Fi(;)202 b Fp(1])403 b Fh(wil)62 b(l)403 b(b)-62 b(e)402 b(fe)-62 b(asible\),)406 b(pr)-62 b(oblem)0 4436 y(\(25\))432 b(do)-62 b(es)432 b(not)h(have)f(a)h(solution.)1882 6562 y Fv(Pro)39 b(of)542 b(of)h(theorem:)675 b Fp(F)-101 b(or)472 b(simplicit)-34 b(y)-101 b(,)488 b(w)-34 b(e)473 b(will)f(assume)g(that)i Fv(\247)33826 6744 y Fo(x)34925 6562 y Fp(is)e(p)34 b(ositiv)-34 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b(in)-34 b(tro)34 b(duce)521 b(the)f(follo)-34 b(wing)521 b(notation.)887 b(Let)0 4869 y Fj(f)p Fv(x)1342 5051 y Fl(i)1717 4869 y Fj(g)2323 4340 y Fl(N)3070 4451 y Fe(x)2323 5230 y Fl(i)p Fm(=1)4217 4869 y Fp(and)316 b Fj(f)p Fv(y)7827 5051 y Fl(i)8202 4869 y Fj(g)8808 4211 y Fl(N)9555 4322 y Fe(y)8808 5230 y Fl(i)p Fm(=1)10701 4869 y Fp(b)34 b(e)315 b(the)h(training)g(data)g(p)34 b(oin)-34 b(ts)316 b(in)f(the)h(classes)f(corresp)34 b(onding)315 b(to)h Fv(x)f Fp(and)h Fv(y)334 b Fp(resp)34 b(ectiv)-34 b(ely)-101 b(,)0 6374 y(and)405 b(let)f Fj(f)p Fv(t)5256 6556 y Fl(i)5632 6374 y Fj(g)6238 5934 y Fl(N)6238 6714 y(i)p Fm(=1)8220 6374 y Fp(denote)g(the)h(set)f(of)h(all)e Fi(N)469 b Fp(=)336 b Fi(N)22865 6556 y Fl(x)23720 6374 y Fp(+)269 b Fi(N)25906 6556 y Fl(y)26862 6374 y Fp(training)404 b(data)h(p)34 b(oin)-34 b(ts)405 b(where)14229 8881 y Fv(t)14771 9063 y Fl(i)16254 8881 y Fp(=)1106 b Fv(x)19039 9063 y Fl(i)24263 8881 y Fi(i)337 b Fp(=)f(1)p Fi(;)202 b Fp(2)p 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y Fl(y)21533 14731 y Fp(ro)-34 b(ws)405 b(of)f Fv(K)g Fp(are)f(denoted)i(b)-34 b(y)405 b Fv(K)36594 14913 y Fo(x)37625 14731 y Fp(and)g Fv(K)41075 14913 y Fo(y)41718 14731 y Fp(,)f(resp)34 b(ectiv)-34 b(ely:)22616 18005 y Fv(K)337 b Fp(=)25325 16296 y Ff(\265)26225 17251 y Fv(K)27318 17433 y Fo(x)26217 18756 y Fv(K)27310 18938 y Fo(y)27953 16296 y Ff(\266)29047 18005 y Fi(:)0 21161 y Fp(The)308 b(blo)34 b(c)-34 b(k-ro)g(w-a)g(v)g(eraged)309 b(Gram)f(matrix)f Fv(L)h Fp(is)g(then)h(obtained)g(b)-34 b(y)308 b(setting)h(the)f(ro)-34 b(w)309 b(a)-34 b(v)g(erage)307 b(of)h(the)h Fv(K)48174 21343 y Fo(x)48801 21161 y Fp(-blo)34 b(c)-34 b(k)0 22667 y(and)405 b(the)f Fv(K)5537 22849 y Fo(y)6180 22667 y Fp(-blo)34 b(c)-34 b(k)404 b(equal)f(to)i(zero:) 16630 25952 y Fv(L)338 b Fp(=)19085 24242 y Ff(\265)19977 25187 y Fv(K)21070 25369 y Fo(x)21967 25187 y Fj(\241)269 b Fv(1)23876 25375 y Fl(N)24623 25486 y Fe(x)25197 25187 y Fv(l)25584 24747 y Fl(T)25584 25486 y Fo(x)19979 26692 y Fv(K)21072 26874 y Fo(y)21984 26692 y Fj(\241)g Fv(1)23893 26880 y Fl(N)24640 26991 y Fe(y)25195 26692 y Fv(l)25582 26252 y Fl(T)25582 26992 y Fo(y)26318 24242 y Ff(\266)27547 25952 y Fp(=)28827 24242 y Ff(\265)29811 24255 y Fj(p)p 30821 24255 1559 49 v 935 x Fi(N)31795 25372 y Fl(x)32380 25190 y Fv(L)33218 25372 y Fo(x)29719 25739 y Ff(p)p 30931 25739 1527 49 v 970 x Fi(N)31905 26891 y Fl(y)32457 26709 y Fv(L)33295 26891 y Fo(y)33939 24242 y Ff(\266)35033 25952 y Fi(;)0 29321 y Fp(where)312 b Fv(1)4110 29503 y Fl(m)5310 29321 y Fp(is)g(a)h(column)g(v)-34 b(ector)312 b(of)g(ones)h(of)g(dimension)g Fi(m)p Fp(.)508 b(The)313 b(ro)-34 b(w)313 b(a)-34 b(v)g(erages)312 b Fv(l)37544 28881 y Fl(T)37544 29621 y Fo(x)38590 29321 y Fp(and)h Fv(l)41242 28881 y Fl(T)41242 29621 y Fo(y)42289 29321 y Fp(are)f Fi(N)132 b Fp(-dimensional)0 30827 y(ro)-34 b(w)405 b(v)-34 b(ectors)403 b(giv)-34 b(en)404 b(b)-34 b(y:)18574 34419 y(\()p Fv(l)19432 33918 y Fl(T)19432 34718 y Fo(x)20166 34419 y Fp(\))20637 34601 y Fl(i)22120 34419 y Fp(=)24779 33599 y(1)p 24303 34140 1559 49 v 24303 35250 a Fi(N)25277 35432 y Fl(x)26439 32901 y(N)27186 33012 y Fe(x)26197 33267 y Ff(X)26255 35873 y Fl(j)51 b Fm(=1)28149 34419 y Fi(K)87 b Fp(\()p Fv(x)30472 34601 y Fl(j)30960 34419 y Fi(;)202 b Fv(t)32041 34601 y Fl(i)32416 34419 y Fp(\))p Fi(;)18574 38983 y Fp(\()p Fv(l)19432 38482 y Fl(T)19432 39282 y Fo(y)20166 38983 y Fp(\))20637 39165 y Fl(i)22120 38983 y Fp(=)24763 38163 y(1)p 24303 38704 1527 49 v 24303 39814 a Fi(N)25277 39996 y Fl(y)26416 37335 y(N)27163 37446 y Fe(y)26164 37831 y Ff(X)26222 40437 y Fl(j)51 b Fm(=1)28117 38983 y Fi(K)87 b Fp(\()p Fv(y)30440 39165 y Fl(j)30927 38983 y Fi(;)202 b Fv(t)32008 39165 y Fl(i)32383 38983 y Fp(\))p Fi(:)0 43283 y Fv(5.3)697 b(Result)0 45428 y(Theorem)464 b(7)606 b Fh(If)519 b Fv(l)8967 45610 y Fo(x)10085 45428 y Fp(=)490 b Fv(l)11905 45610 y Fo(y)12548 45428 y Fh(,)539 b(then)518 b(the)f(minimax)h(pr)-62 b(ob)g(ability)515 b(de)-62 b(cision)516 b(pr)-62 b(oblem)517 b(\(31\))f(do)-62 b(es)517 b(not)h(have)e(a)i(so-)0 46934 y(lution.)725 b(Otherwise,)502 b(an)490 b(optimal)e(de)-62 b(cision)488 b(b)-62 b(oundary)488 b Fj(D)34 b Fp(\()p Fi(\260)67 b(;)202 b(b)p Fp(\))490 b Fh(c)-62 b(an)490 b(b)-62 b(e)489 b(determine)-62 b(d)488 b(by)h(solving)f(the)h(c)-62 b(onvex)0 48439 y(optimization)431 b(pr)-62 b(oblem:)9523 51368 y Fi(\267)10221 50867 y Fn(\241)p Fm(1)10221 51667 y Fn(\244)11816 51368 y Fp(:=)336 b(min)14174 52095 y Fl(\260)15655 49884 y Ff(q)p 16867 49884 5163 49 v 1484 x Fi(\260)17562 51018 y Fl(T)18295 51368 y Fv(L)19133 51018 y Fl(T)19133 51667 y Fo(x)19868 51368 y Fv(L)20706 51550 y Fo(x)21334 51368 y Fi(\260)g Fp(+)23510 49970 y Ff(q)p 24722 49970 5178 49 v 1398 x Fi(\260)25417 51018 y Fl(T)26151 51368 y Fv(L)26989 51018 y Fl(T)26989 51667 y Fo(y)27723 51368 y Fv(L)28561 51550 y Fo(y)29205 51368 y Fi(\260)1306 b Fh(s.t.)1237 b Fi(\260)34714 50867 y Fl(T)35448 51368 y Fp(\()p Fv(l)36306 51550 y Fo(x)37204 51368 y Fj(\241)269 b Fv(l)38803 51550 y Fo(y)39446 51368 y Fp(\))337 b(=)g(1)p Fi(;)7368 b Fp(\(32\))0 54342 y Fh(and)433 b(setting)f Fi(b)i Fh(to)f(the)f(value)18509 56270 y Fi(b)19029 56452 y Fn(\244)19892 56270 y Fp(=)337 b Fi(\260)21867 55769 y Fl(T)21800 56569 y Fn(\244)22600 56270 y Fv(l)22987 56452 y Fo(x)23885 56270 y Fj(\241)269 b Fi(\267)25795 56452 y Fn(\244)26321 54786 y Ff(q)p 27533 54786 5621 49 v 1484 x Fi(\260)28228 55919 y Fl(T)28161 56569 y Fn(\244)28962 56270 y Fv(L)29800 55919 y Fl(T)29800 56569 y Fo(x)30534 56270 y Fv(L)31372 56452 y Fo(x)32000 56270 y Fi(\260)32628 56452 y Fn(\244)33154 56270 y Fi(;)0 58550 y Fh(wher)-62 b(e)565 b Fi(\260)4183 58732 y Fn(\244)5274 58550 y Fh(is)g(an)h(optimal)e(solution)f(of)i(\(32\).)952 b(The)565 b(optimal)f(worst-c)-62 b(ase)563 b(misclassi\257c)-62 b(ation)563 b(pr)-62 b(ob)g(ability)562 b(is)0 60056 y(obtaine)-62 b(d)431 b(via)10988 63089 y Fp(1)269 b Fj(\241)h Fi(\256)13851 63271 y Fn(\244)14713 63089 y Fp(=)17479 62269 y(1)p 16126 62811 3312 49 v 16126 63921 a(1)f(+)g Fi(\267)18911 63571 y Fm(2)18911 64221 y Fn(\244)19907 63089 y Fp(=)22363 60477 y Ff(\263)23087 60761 y(p)p 24299 60761 5621 49 v 1061 x Fi(\260)24994 61472 y Fl(T)24927 62122 y Fn(\244)25728 61822 y Fv(L)26566 61472 y Fl(T)26566 62122 y Fo(x)27300 61822 y Fv(L)28138 62004 y Fo(x)28766 61822 y Fi(\260)29394 62004 y Fn(\244)30189 61822 y Fp(+)31401 60483 y Ff(q)p 32613 60483 5636 49 v 1339 x Fi(\260)33308 61472 y Fl(T)33241 62122 y Fn(\244)34042 61822 y Fv(L)34880 61472 y Fl(T)34880 62122 y Fo(y)35614 61822 y Fv(L)36452 62004 y Fo(y)37096 61822 y Fi(\260)37724 62004 y Fn(\244)38249 60477 y Ff(\264)38973 60737 y Fm(2)p 21319 62811 19224 49 v 21319 64612 a Fp(1)h(+)23407 63266 y Ff(\263)24131 63551 y(p)p 25343 63551 5621 49 v 1061 x Fi(\260)26038 64262 y Fl(T)25971 64912 y Fn(\244)26772 64612 y Fv(L)27610 64262 y Fl(T)27610 64912 y Fo(x)28344 64612 y Fv(L)29182 64794 y Fo(x)29810 64612 y Fi(\260)30438 64794 y Fn(\244)31233 64612 y Fp(+)32445 63273 y Ff(q)p 33657 63273 5636 49 v 1339 x Fi(\260)34352 64262 y Fl(T)34285 64912 y Fn(\244)35086 64612 y Fv(L)35924 64262 y Fl(T)35924 64912 y Fo(y)36658 64612 y Fv(L)37496 64794 y Fo(y)38139 64612 y Fi(\260)38767 64794 y Fn(\244)39293 63266 y Ff(\264)40017 63527 y Fm(2)40675 63089 y Fi(:)0 67174 y Fh(Be)-62 b(c)g(ause)394 b(b)-62 b(oth)395 b Fv(L)7936 66734 y Fl(T)7936 67473 y Fo(x)8670 67174 y Fv(L)9508 67356 y Fo(x)10533 67174 y Fh(and)i Fv(L)13689 66734 y Fl(T)13689 67473 y Fo(y)14423 67174 y Fv(L)15261 67356 y Fo(y)16301 67174 y Fh(ar)-62 b(e)396 b(p)-62 b(ositive)393 b(semide\257nite)i(but)g(not)i(p)-62 b(ositive)393 b(de\257nite,)403 b(the)395 b(optimal)g(hyp)-62 b(er-)0 68679 y(plane)433 b(is)g(not)g(unique.)25394 72000 y Fp(20)p eop %%Page: 21 21 21 20 bop 1882 1218 a Fp(Once)392 b(an)h(optimal)g(decision)g(b)34 b(oundary)394 b(is)e(found,)k(classi\257cation)d(of)g(a)g(new)g(data)h (p)34 b(oin)-34 b(t)394 b Fv(z)44475 1400 y Fl(new)46629 1218 y Fp(is)e(done)i(b)-34 b(y)0 2958 y(ev)-67 b(aluating)419 b(sign)q(\()p Fv(a)9084 2518 y Fl(T)9084 3258 y Fn(\244)9818 2958 y Fi(')p Fp(\()p Fv(z)11702 3140 y Fl(new)13463 2958 y Fp(\))280 b Fj(\241)g Fi(b)15957 3140 y Fn(\244)16483 2958 y Fp(\))363 b(=)f(sign)20919 1613 y Ff(\263\263)22366 2049 y(P)23646 2301 y Fl(N)24393 2412 y Fe(x)24912 2301 y Fm(+)p Fl(N)26391 2412 y Fe(y)23646 3320 y Fl(i)p Fm(=1)26945 2958 y Fp([)p Fi(\260)27910 3140 y Fn(\244)28435 2958 y Fp(])28772 3140 y Fl(i)29148 2958 y Fi(K)87 b Fp(\()p Fv(t)31277 3140 y Fl(i)31653 2958 y Fi(;)202 b Fv(z)32812 3140 y Fl(new)34573 2958 y Fp(\))35044 1613 y Ff(\264)36038 2958 y Fj(\241)269 b Fi(b)37770 3140 y Fn(\244)38296 1613 y Ff(\264)39439 2958 y Fp(\(notice)420 b(that)h(this)f(can)g(b)34 b(e)0 4766 y(ev)-67 b(aluated)380 b(only)g(in)h(terms)f(of)h(the)g(k) -34 b(ernel)379 b(function\);)390 b(if)380 b(this)h(is)f(+1,)385 b Fv(z)32799 4948 y Fl(new)34940 4766 y Fp(is)380 b(classi\257ed)h(as)f (b)34 b(elonging)381 b(to)f(class)0 6271 y Fv(x)p Fp(,)403 b(otherwise)i Fv(z)7494 6453 y Fl(new)9659 6271 y Fp(is)f(classi\257ed) g(as)g(b)34 b(elonging)404 b(to)h(class)f Fv(y)19 b Fp(.)1882 9295 y Fv(Pro)39 b(of)546 b(of)g(theorem:)681 b Fp(In)475 b(order)f(to)i(solv)-34 b(e)474 b(\(31\),)494 b(w)-34 b(e)475 b(can)h(apply)f(Theorem)g(2.)751 b(If)476 b(the)f (distributions)i(of)0 10801 y Fi(')p Fp(\()p Fv(x)p Fp(\))402 b(and)g Fi(')p Fp(\()p Fv(y)19 b Fp(\))402 b(ha)-34 b(v)g(e)402 b(the)g(same)g(mean:)p 19733 9746 2472 49 v 537 w Fi(')p Fp(\()p Fv(x)p Fp(\))337 b(=)p 23821 9746 2491 49 v 337 w Fi(')p Fp(\()p Fv(y)19 b Fp(\),)402 b(then)g(the)g(minimax)g (probabilit)-34 b(y)402 b(decision)f(problem)0 12306 y(\(31\))407 b(do)34 b(es)407 b(not)g(ha)-34 b(v)g(e)407 b(a)f(solution.)547 b(Otherwise,)406 b(an)h(optimal)g(h)-34 b(yp)34 b(erplane)407 b Fj(H)12 b Fp(\()p Fv(a)p Fi(;)202 b(b)p Fp(\))406 b(in)g Fg(R)41182 11866 y Fl(f)42194 12306 y Fp(can)h(b)34 b(e)406 b(determined)0 13812 y(b)-34 b(y)405 b(solving)e(the)i(con)-34 b(v)g(ex)404 b(optimization)h (problem:)8440 16912 y Fi(\267)9138 16412 y Fn(\241)p Fm(1)9138 17212 y Fn(\244)10733 16912 y Fp(:=)336 b(min)13095 17639 y Fo(a)14571 15512 y Ff(q)p 15783 15512 4938 49 v 1400 x Fv(a)16461 16562 y Fl(T)17195 16912 y Fp(\247)18070 17159 y Fl(')p Fm(\()p Fo(x)p Fm(\))20043 16912 y Fv(a)269 b Fp(+)22202 15512 y Ff(q)p 23414 15512 4953 49 v 1400 x Fv(a)24092 16562 y Fl(T)24826 16912 y Fp(\247)25701 17159 y Fl(')p Fm(\()p Fo(y)15 b Fm(\))27689 16912 y Fv(a)1106 b Fp(s.t.)h Fv(a)32881 16412 y Fl(T)33615 16912 y Fp(\()p 34086 15858 2472 49 v Fi(')p Fp(\()p Fv(x)p Fp(\))270 b Fj(\241)p 38039 15858 2491 49 v 269 w Fi(')p Fp(\()p Fv(y)19 b Fp(\))q(\))337 b(=)f(1)p Fi(;)6285 b Fp(\(33\))0 19970 y(and)405 b(setting)g Fi(b)f Fp(to)g(the)h(v)-67 b(alue)17869 21853 y Fi(b)18389 22035 y Fn(\244)19251 21853 y Fp(=)337 b Fv(a)21209 21353 y Fl(T)21209 22153 y Fn(\244)p 21942 20799 2472 49 v 21942 21853 a Fi(')p Fp(\()p Fv(x)p Fp(\))270 b Fj(\241)f Fi(\267)26593 22035 y Fn(\244)27119 20454 y Ff(q)p 28331 20454 5464 49 v 1399 x Fv(a)29009 21503 y Fl(T)29009 22153 y Fn(\244)29743 21853 y Fp(\247)30618 22100 y Fl(')p Fm(\()p Fo(x)p Fm(\))32591 21853 y Fv(a)33269 22035 y Fn(\244)33795 21853 y Fi(;)15713 b Fp(\(34\))0 24372 y(where)404 b Fv(a)4183 24554 y Fn(\244)5113 24372 y Fp(is)f(an)i(optimal)f(solution)h(of)g(\(33\).)1882 25891 y(Ho)-34 b(w)g(ev)g(er,)514 b(w)-34 b(e)492 b(do)g(not)h(wish)f (to)h(solv)-34 b(e)491 b(the)i(con)-34 b(v)g(ex)491 b(optimization)i (problem)f(in)g(this)g(form,)514 b(b)34 b(ecause)492 b(w)-34 b(e)0 27397 y(w)g(an)g(t)372 b(to)f(a)-34 b(v)g(oid)370 b(using)h Fi(f)501 b Fp(or)370 b Fi(')g Fp(explicitly)-101 b(.)526 b(W)-101 b(e)369 b(\257rst)i(form)g(sample)f(estimates)g(of)h (the)f(means)h(and)g(co)-34 b(v)-67 b(ariances)p 0 27848 2472 49 v 0 28902 a Fi(')p Fp(\()p Fv(x)p Fp(\))p Fi(;)p 3010 27848 2491 49 v 202 w(')p Fp(\()p Fv(y)19 b Fp(\))q Fi(;)202 b Fp(\247)6915 29149 y Fl(')p Fm(\()p Fo(x)p Fm(\))9292 28902 y Fp(and)405 b(\247)12524 29149 y Fl(')p Fm(\()p Fo(y)15 b Fm(\))14512 28902 y Fp(:)13271 32635 y Fg([)13180 32983 y Fi(')p Fp(\()p Fv(x)p Fp(\))1107 b(=)19417 32163 y(1)p 18941 32704 1559 49 v 18941 33814 a Fi(N)19915 33996 y Fl(x)21077 31465 y(N)21824 31576 y Fe(x)20835 31831 y Ff(X)20949 34437 y Fl(i)p Fm(=1)22787 32983 y Fi(')p Fp(\()p Fv(x)24787 33165 y Fl(i)25163 32983 y Fp(\))p Fi(;)13261 37026 y Fg([)13161 37374 y Fi(')p Fp(\()p Fv(y)19 b Fp(\))1107 b(=)19401 36554 y(1)p 18941 37096 1527 49 v 18941 38206 a Fi(N)19915 38388 y Fl(y)21054 35727 y(N)21801 35838 y Fe(y)20802 36223 y Ff(X)20916 38829 y Fl(i)p Fm(=1)22755 37374 y Fi(')p Fp(\()p Fv(y)24755 37556 y Fl(i)25130 37374 y Fp(\))p Fi(;)12872 41327 y Fp(^)12672 41637 y Fv(\247)13679 41884 y Fl(')p Fm(\()p Fo(x)p Fm(\))16758 41637 y Fp(=)19417 40817 y(1)p 18941 41358 1559 49 v 18941 42468 a Fi(N)19915 42650 y Fl(x)21077 40119 y(N)21824 40230 y Fe(x)20835 40485 y Ff(X)20949 43091 y Fl(i)p Fm(=1)22585 41637 y Fp(\()p Fi(')p Fp(\()p Fv(x)25056 41819 y Fl(i)25433 41637 y Fp(\))269 b Fj(\241)27476 41289 y Fg([)27385 41637 y Fi(')p Fp(\()p Fv(x)p Fp(\))q(\)\()p Fi(')p Fp(\()p Fv(x)32799 41819 y Fl(i)33175 41637 y Fp(\))h Fj(\241)35219 41289 y Fg([)35128 41637 y Fi(')p Fp(\()p Fv(x)p Fp(\)\))38070 41137 y Fl(T)38805 41637 y Fi(;)12857 45719 y Fp(^)12657 46029 y Fv(\247)13664 46275 y Fl(')p Fm(\()p Fo(y)15 b Fm(\))16758 46029 y Fp(=)19401 45209 y(1)p 18941 45750 1527 49 v 18941 46860 a Fi(N)19915 47042 y Fl(y)21054 44381 y(N)21801 44492 y Fe(y)20802 44877 y Ff(X)20916 47483 y Fl(i)p Fm(=1)22553 46029 y Fp(\()p Fi(')p Fp(\()p Fv(y)25024 46211 y Fl(i)25400 46029 y Fp(\))269 b Fj(\241)27453 45681 y Fg([)27353 46029 y Fi(')p Fp(\()p Fv(y)19 b Fp(\)\)\()p Fi(')p Fp(\()p Fv(y)32785 46211 y Fl(i)33162 46029 y Fp(\))269 b Fj(\241)35215 45681 y Fg([)35115 46029 y Fi(')p Fp(\()p Fv(y)19 b Fp(\)\))38076 45528 y Fl(T)38811 46029 y Fi(;)0 49881 y Fp(and)405 b(plug)g(these)f(in)-34 b(to)405 b(the)f(ob)67 b(jectiv)-34 b(e)405 b(function)g(and)g(the)g (constrain)-34 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b(expressed)f(in)i(terms)f(of)g(inner)g(pro)34 b(ducts)441 b(of)g Fi(')p Fp(\()p Fv(z)p Fp(\))f(only)-101 b(.)647 b(When)440 b(substituting)j(expression)d(\(36\))h(for)25394 72000 y(21)p eop %%Page: 22 22 22 21 bop 0 1218 a Fv(a)388 b Fp(in)g(\(35\),)k(w)-34 b(e)389 b(indeed)f(see)g(that)i(b)34 b(oth)389 b(the)g(ob)67 b(jectiv)-34 b(e)388 b(and)h(the)g(constrain)-34 b(ts)390 b(can)e(b)34 b(e)388 b(written)h(in)g(terms)f(of)g(the)0 2723 y(k)-34 b(ernel)403 b(function)j Fi(K)87 b Fp(\()p Fv(z)10527 2905 y Fm(1)11053 2723 y Fi(;)202 b Fv(z)12212 2905 y Fm(2)12737 2723 y Fp(\))337 b(=)f Fi(')p Fp(\()p Fv(z)16708 2905 y Fm(1)17234 2723 y Fp(\))17705 2283 y Fl(T)18439 2723 y Fi(')p Fp(\()p Fv(z)20323 2905 y Fm(2)20849 2723 y Fp(\).)539 b(W)-101 b(e)403 b(obtain:)9665 5920 y Fi(\267)10363 5419 y Fn(\241)p Fm(1)10363 6220 y Fn(\244)11958 5920 y Fp(:=)336 b(min)14316 6647 y Fl(\260)15796 4436 y Ff(q)p 17008 4436 5163 49 v 1484 x Fi(\260)17703 5570 y Fl(T)18437 5920 y Fv(L)19275 5570 y Fl(T)19275 6220 y Fo(x)20009 5920 y Fv(L)20847 6102 y Fo(x)21475 5920 y Fi(\260)h Fp(+)23652 4522 y Ff(q)p 24864 4522 5178 49 v 1398 x Fi(\260)25559 5570 y Fl(T)26293 5920 y Fv(L)27131 5570 y Fl(T)27131 6220 y Fo(y)27865 5920 y Fv(L)28703 6102 y Fo(y)29346 5920 y Fi(\260)1174 b Fp(s.t.)1107 b Fi(\260)34573 5419 y Fl(T)35307 5920 y Fp(\()p Fv(l)36165 6102 y Fo(x)37062 5920 y Fj(\241)270 b Fv(l)38662 6102 y Fo(y)39305 5920 y Fp(\))337 b(=)f(1)p Fi(;)0 9298 y Fp(where)404 b Fi(\260)g Fp(=)336 b([)p Fi(\256)6928 9480 y Fm(1)7858 9298 y Fi(\256)8633 9480 y Fm(2)9765 9298 y Fj(\242)202 b(\242)g(\242)606 b Fi(\256)12561 9486 y Fl(N)13308 9597 y Fe(x)14286 9298 y Fi(\257)14972 9480 y Fm(1)15901 9298 y Fi(\257)16587 9480 y Fm(2)17719 9298 y Fj(\242)202 b(\242)g(\242)605 b Fi(\257)20425 9486 y Fl(N)21172 9597 y Fe(y)21727 9298 y Fp(])22064 8859 y Fl(T)22797 9298 y Fp(.)539 b(W)-101 b(e)403 b(can)h(also)g (write)g(this)h(as)12140 12144 y Fi(\267)12838 11644 y Fn(\241)p Fm(1)12838 12444 y Fn(\244)14433 12144 y Fp(:=)336 b(min)16791 12872 y Fl(\260)18271 12144 y Fj(k)p Fv(L)19715 12326 y Fo(x)20344 12144 y Fi(\260)67 b Fj(k)21645 12326 y Fm(2)22440 12144 y Fp(+)269 b Fj(k)p Fv(L)25096 12326 y Fo(y)25739 12144 y Fi(\260)67 b Fj(k)27040 12326 y Fm(2)28673 12144 y Fp(s.t.)1107 b Fi(\260)32098 11644 y Fl(T)32831 12144 y Fp(\()p Fv(l)33689 12326 y Fo(x)34587 12144 y Fj(\241)269 b Fv(l)36186 12326 y Fo(y)36830 12144 y Fp(\))337 b(=)f(1)p Fi(;)0 15387 y Fp(whic)-34 b(h)401 b(is)e(a)h(second)g(order)g(cone)f(program)h(\(SOCP\))h(\(Bo)-34 b(yd)401 b(and)f(V)-101 b(anden)-34 b(b)34 b(erghe,)401 b(2001\))f(that)h(has)g(the)f(same)0 16892 y(form)447 b(as)g(the)g(SOCP)g(in)g(\(12\))h(and)f(can)g(th)-34 b(us)448 b(b)34 b(e)447 b(solv)-34 b(ed)447 b(in)f(a)h(similar)f(w)-34 b(a)g(y)-101 b(.)667 b(Notice)446 b(that,)459 b(in)447 b(this)g(case,)457 b(the)0 18398 y(optimizing)519 b(v)-67 b(ariable)517 b(is)h Fi(\260)594 b Fj(2)527 b Fg(R)15536 17958 y Fl(N)16283 18069 y Fe(x)16802 17958 y 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46379 y Fl(i)24190 46197 y Fp(\()p Fi(')p Fp(\()p Fv(x)26661 46379 y Fl(i)27037 46197 y Fp(\))270 b Fj(\241)p 28990 45143 2472 49 v 269 w Fi(')p Fp(\()p Fv(x)p Fp(\)\)\()p Fi(')p Fp(\()p Fv(x)34403 46379 y Fl(i)34780 46197 y Fp(\))f Fj(\241)p 36732 45143 V 269 w Fi(')p Fp(\()p Fv(x)p Fp(\))q(\))39675 45697 y Fl(T)40409 46197 y Fi(;)11052 50589 y Fv(\247)12059 50836 y Fl(')p Fm(\()p Fo(x)p Fm(\))15139 50589 y Fp(=)1107 b Fi(\275)17816 50771 y Fl(y)18367 50589 y Fi(I)18900 50786 y Fl(f)19775 50589 y Fp(+)21239 48942 y Fl(N)21986 49053 y Fe(y)20987 49438 y Ff(X)21101 52043 y Fl(i)p Fm(=1)22940 50589 y Fp(\255)23815 50771 y Fl(i)24191 50589 y Fp(\()p Fi(')p Fp(\()p Fv(y)26662 50771 y Fl(i)27038 50589 y Fp(\))270 b Fj(\241)p 28991 49534 2491 49 v 269 w Fi(')p Fp(\()p Fv(y)19 b Fp(\)\)\()p Fi(')p Fp(\()p Fv(y)34423 50771 y Fl(i)34800 50589 y Fp(\))270 b Fj(\241)p 36753 49534 V 269 w Fi(')p Fp(\()p Fv(y)19 b Fp(\)\))39714 50089 y Fl(T)40449 50589 y Fi(;)0 54319 y Fp(where)339 b Fi(I)3973 54516 y Fl(f)4917 54319 y Fp(is)g(the)g(iden)-34 b(tit)g(y)340 b(matrix)f(of)g(dimension)h Fi(f)130 b Fp(.)517 b(Indeed,)352 b(from)340 b(Corollary)e(6,)352 b(w)-34 b(e)339 b(kno)-34 b(w)340 b(that)g(the)g(optimal)0 55824 y Fv(a)313 b Fp(will)g(lie)g(in)h(the)g(span)h(of)f(the)g(mapp)34 b(ed)314 b(data)h Fj(f)p Fi(')p Fp(\()p Fv(t)23674 56006 y Fl(i)24050 55824 y Fp(\))p Fj(g)25127 55385 y Fl(N)25127 56164 y(i)p Fm(=1)27019 55824 y Fp(and)g(as)f(a)f(consequence,)331 b(the)315 b(optimization)f(problem)0 57330 y(\(31\))371 b(can)g(b)34 b(e)370 b(en)-34 b(tirely)369 b(expressed)h(in)h(terms)f (of)g(inner)h(pro)34 b(ducts)371 b(of)g(the)f(mapp)34 b(ed)371 b(data)g(only)-101 b(.)527 b(Ev)-34 b(en)371 b(if)f(w)-34 b(e)371 b(tak)-34 b(e)0 58835 y(uncertain)g(t)g(y)498 b(in)f(the)g(means)g(and)h(co)-34 b(v)-67 b(ariance)495 b(matrices)h(in)-34 b(to)498 b(accoun)-34 b(t)498 b(\(that)g(is)f Fi(\272)568 b Fj(6)p Fp(=)491 b(0)p Fi(;)202 b(\275)43369 59017 y Fl(x)44444 58835 y Fp(=)491 b Fi(\275)46505 59017 y Fl(y)47548 58835 y Fj(6)p Fp(=)g(0,)519 b(see)0 60341 y(Section)404 b(3\),)h(w)-34 b(e)404 b(can)g(still)g(k)-34 b(ernelize)403 b(the)h(MPM)g(and)h(obtain)g(the)g(con)-34 b(v)g(ex)404 b(optimization)h(problem)4848 63536 y Fi(\267)5546 63035 y Fn(\241)p Fm(1)5546 63836 y Fn(\244)7140 63536 y Fp(:=)337 b(min)9498 64263 y Fl(\260)10979 62054 y Ff(q)p 12191 62054 9891 49 v 1482 x Fi(\260)12886 63186 y Fl(T)13620 63536 y Fp(\()p Fv(L)14929 63186 y Fl(T)14929 63836 y Fo(x)15663 63536 y Fv(L)16501 63718 y Fo(x)17399 63536 y Fp(+)269 b Fi(\275)19238 63718 y Fl(x)19823 63536 y Fv(K)p Fp(\))p Fi(\260)336 b Fp(+)23563 62138 y Ff(q)p 24775 62138 9873 49 v 1398 x Fi(\260)25470 63186 y Fl(T)26204 63536 y Fp(\()p Fv(L)27513 63186 y Fl(T)27513 63836 y Fo(y)28247 63536 y Fv(L)29085 63718 y Fo(y)29998 63536 y Fp(+)269 b Fi(\275)31837 63718 y Fl(y)32389 63536 y Fv(K)p Fp(\))p Fi(\260)1279 b Fp(s.t.)1212 b Fi(\260)39390 63035 y Fl(T)40124 63536 y Fp(\()p Fv(l)40982 63718 y Fo(x)41880 63536 y Fj(\241)269 b Fv(l)43479 63718 y Fo(y)44122 63536 y Fp(\))337 b(=)g(1)p Fi(;)0 66711 y Fp(and)405 b(optimal)f(v)-67 b(alue)404 b(for)g Fi(b)16145 68637 y(b)16665 68819 y Fn(\244)17528 68637 y Fp(=)336 b Fi(\260)19502 68136 y Fl(T)19435 68937 y Fn(\244)20236 68637 y Fv(l)20623 68819 y Fo(x)21521 68637 y Fj(\241)269 b Fi(\267)23431 68819 y Fn(\244)23957 67155 y Ff(q)p 25169 67155 10350 49 v 1482 x Fi(\260)25864 68287 y Fl(T)25797 68937 y Fn(\244)26598 68637 y Fp(\()p Fv(L)27907 68287 y Fl(T)27907 68937 y Fo(x)28641 68637 y Fv(L)29479 68819 y Fo(x)30377 68637 y Fp(+)g Fi(\275)32216 68819 y Fl(x)32801 68637 y Fv(K)p Fp(\))p Fi(\260)34993 68819 y Fn(\244)35518 68637 y Fi(:)25394 72000 y Fp(22)p eop %%Page: 23 23 23 22 bop 0 1218 a Fp(The)405 b(optimal)f(w)-34 b(orst-case)405 b(misclassi\257cation)f(probabilit)-34 b(y)405 b(is)e(obtained)j(via) 17109 4607 y(1)269 b Fj(\241)g Fi(\256)19975 4106 y Fm(rob)19971 4907 y Fn(\244)21728 4607 y Fp(=)28478 3787 y(1)p 23140 4328 11282 49 v 23140 5438 a(1)h(+)f(max\(0)p Fi(;)202 b Fp(\()p Fi(\267)30269 5620 y Fn(\244)31065 5438 y Fj(\241)269 b Fi(\272)77 b Fp(\)\))33895 5088 y Fm(2)34554 4607 y Fi(:)1882 8210 y Fp(This)522 b(approac)-34 b(h)523 b(is)e(able)g(to)h (deal)f(with)i(uncertain)-34 b(t)g(y)523 b(in)e(the)h(means)g(and)g(co) -34 b(v)-67 b(ariance)521 b(matrices)g(of)h(the)0 9716 y(mapp)34 b(ed)418 b(data.)579 b(Ho)-34 b(w)g(ev)g(er,)420 b(it)e(will)e(not)i(necessarily)e(regularize)g(the)i(k)-34 b(ernel)416 b(MPM,)h(since)g(the)h(Gram)f(matrix)0 11221 y Fv(K)381 b Fp(ma)-34 b(y)381 b(not)h(b)34 b(e)381 b(p)34 b(ositiv)-34 b(e)381 b(de\257nite.)532 b(F)-101 b(or)381 b(regularization)g(purp)34 b(oses,)386 b(it)381 b(is)g(th)-34 b(us)383 b(adv)-67 b(an)-34 b(tageous)383 b(to)f(add)g(a)f(term)0 12727 y Fi(\275I)1160 12915 y Fl(N)2460 12727 y Fp(to)405 b Fv(K)p Fp(,)e(where)h Fi(\275)336 b(>)h Fp(0)404 b(is)g(small.)0 15907 y Fv(5.4)697 b(Single)466 b(Class)e(Case)0 18127 y Fp(In)418 b(a)g(similar)f(w)-34 b(a)g(y)-101 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y(partitioned)378 b(in)-34 b(to)378 b(90\045)g(training)f(and)i (10\045)e(test)h(sets.)530 b(The)377 b(k)-34 b(ernel)377 b(parameter)g(\()p Fi(\276)43 b Fp(\))378 b(for)g(the)f(Gaussian)i(k) -34 b(ernel)0 17072 y(\()p Fi(e)1035 16632 y Fn(\241k)p Fl(x)p Fn(\241)p Fl(y)32 b Fn(k)4466 16320 y Fc(2)4928 16632 y Fl(=\276)6023 17072 y Fp(\))463 b(w)-34 b(as)463 b(tuned)g(using)g(cross-v)-67 b(alidation)461 b(o)-34 b(v)g(er)462 b(20)g(random)h(partitions)g(of)f(the)h(training)f(set.) 712 b(The)0 18578 y(rep)34 b(orted)560 b(results)g(are)g(the)g(a)-34 b(v)g(erages)560 b(o)-34 b(v)g(er)560 b(50)g(random)h(partitions)g(for) f(b)34 b(oth)561 b(the)g(linear)e(k)-34 b(ernel)560 b(and)g(the)0 20083 y(Gaussian)405 b(k)-34 b(ernel)403 b(with)i Fi(\276)448 b Fp(c)-34 b(hosen)404 b(as)h(ab)34 b(o)-34 b(v)g(e.)p 4403 21293 43195 45 v 5067 22347 a(Dataset)5472 b Fh(Line)-62 b(ar)434 b(kernel)2036 b(Gaussian)432 b(kernel)1847 b(BPB)1919 b(SVML)1399 b(SVMG)p 13263 22843 34334 45 v 15322 23852 a 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b(\045)2918 b(-)2910 b(75.9)404 b(\045)1348 b(86.7)404 b(\045)p 4403 32097 V 0 34445 a(T)-101 b(able)404 b(2:)606 b(Lo)-34 b(w)g(er)413 b(b)34 b(ound)413 b Fi(\256)k Fp(and)c(test-set)g(accuracy)e(\(TSA\))j(compared)f(to)f(BPB)g(\(b)34 b(est)412 b(p)34 b(erformance)412 b(in)h(the)4882 35950 y(article)386 b(of)h(Breiman)f(\(1998\)\))i(and)g(to)f(the)g(p)34 b(erformance)387 b(of)g(an)g(SVM)g(with)h(linear)e(k)-34 b(ernel)386 b(\(SVML\))4882 37456 y(and)405 b(an)g(SVM)f(with)h (Gaussian)g(k)-34 b(ernel)404 b(\(SVMG\).)1882 43086 y(The)343 b(results)g(are)f(comparable)g(with)h(those)h(in)e(the)h (existing)f(literature)h(\(Breiman,)e(1998\))j(and)f(with)g(those)0 44592 y(obtained)335 b(with)f(supp)34 b(ort)335 b(v)-34 b(ector)333 b(mac)-34 b(hines.)516 b(Also,)347 b(w)-34 b(e)334 b(notice)g(that)h Fi(\256)j Fp(is)c(smaller)e(than)j(the)f (test-set)h(accuracy)0 46097 y(in)f(all)g(cases.)515 b(F)-101 b(urthermore,)348 b Fi(\256)339 b Fp(is)334 b(smaller)g(for)g(a)h(linear)e(decision)h(b)34 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49 v 23524 58005 a Fv(a)24202 57655 y Fl(T)24936 58005 y Fv(\247)25943 58187 y Fo(x)26570 58005 y Fv(a)p 23524 58546 3724 49 v 24745 59657 a Fp(4)p Fi(\272)27381 58825 y(;)22127 b Fp(\(40\))0 62150 y(where)405 b(\(38\))g(is)g(substituted)i(to)e(obtain)h(\(39\).)541 b(Substitution)407 b(of)e(\(40\))h(in)e(\(38\))i(giv)-34 b(es)404 b(the)h(optimal)h(v)-67 b(alue)403 b(for)51329 62133 y(\271)51264 62150 y Fv(x)0 63655 y Fp(and)i(leads)f(to:)17787 65379 y(min)12564 66331 y Fm(\271)12513 66343 y Fo(x)g Fm(:)g(\()14572 66331 y(\271)14521 66343 y Fo(x)o Fn(\241)15876 66331 y Fm(\271)15825 66343 y Fo(x)16398 66091 y Fc(0)16859 66343 y Fm(\))17225 66091 y Fe(T)17874 66343 y Fo(\247)18657 66454 y Fk(x)19198 66009 y Fb(\241)p Fc(1)20299 66343 y Fm(\()20716 66331 y(\271)20665 66343 y Fo(x)o Fn(\241)22020 66331 y Fm(\271)21969 66343 y Fo(x)22542 66091 y Fc(0)23003 66343 y Fm(\))p Fn(\267)p Fl(\272)24622 66091 y Fc(2)25621 65379 y Fv(a)26299 64878 y Fl(T)27097 65362 y Fp(\271)27032 65379 y Fv(x)337 b Fp(=)f Fv(a)30062 64878 y Fl(T)30860 65362 y Fp(\271)30796 65379 y Fv(x)31532 64878 y Fm(0)32326 65379 y Fj(\241)270 b Fi(\272)34215 64200 y Ff(p)p 35427 64200 V 1179 x Fv(a)36105 65029 y Fl(T)36838 65379 y Fv(\247)37845 65561 y Fo(x)38473 65379 y Fv(a)o Fi(;)0 68679 y Fp(whic)-34 b(h)405 b(pro)-34 b(v)g(es)404 b(expression)g (\(21\).)25394 72000 y(27)p eop %%Page: 28 28 28 27 bop 0 1218 a Fr(App)42 b(endix)499 b(C.)648 b(Pro)42 b(of)498 b(of)g(Expression)h(\(23\))0 3532 y Fp(T)-101 b(o)405 b(solv)-34 b(e)403 b(the)i(constrained)g(optimization)f (problem)17882 6283 y(max)18619 7092 y Fo(\247)20610 6283 y Fv(a)21288 5782 y Fl(T)22021 6283 y Fv(\247a)740 b Fp(:)h Fj(k)p Fv(\247)269 b Fj(\241)g Fv(\247)29625 5782 y Fm(0)30150 6283 y Fj(k)30756 6471 y Fl(F)31875 6283 y Fj(\267)336 b Fi(\275;)0 9580 y Fp(w)-34 b(e)393 b(can)g(let)f Fv(\247)h Fp(b)34 b(e)392 b(of)h(the)g(form)g Fv(\247)337 b Fp(=)f Fv(\247)18690 9140 y Fm(0)19462 9580 y Fp(+)246 b Fi(\275)p Fp(\242)p Fv(\247)p Fp(,)394 b(without)h(loss)d(of)i(generalit)-34 b(y)-101 b(.)534 b(Recall)391 b(that)j(the)f(norm)g(used)0 11085 y(here)404 b(is)f(the)i(F)-101 b(rob)34 b(enius)405 b(norm:)538 b Fj(k)p Fi(A)p Fj(k)17279 10645 y Fm(2)17279 11445 y Fl(F)18398 11085 y Fp(=)336 b Fv(T)-116 b(r)p Fp(\()p Fi(A)22485 10645 y Fl(T)23219 11085 y Fi(A)p Fp(\).)539 b(The)405 b(optimization)f(problem)g(then)h(b)34 b(ecomes:)19685 13836 y(max)17300 14702 y Fm(\242)p Fo(\247)405 b Fm(:)f Fn(k)p Fm(\242)p Fo(\247)p Fn(k)22444 14851 y Fe(F)23123 14702 y Fn(\267)p Fm(1)24528 13836 y Fv(a)25206 13335 y Fl(T)25939 13836 y Fv(\247)26946 13335 y Fm(0)27472 13836 y Fv(a)269 b Fp(+)g Fi(\275)p Fv(a)30936 13335 y Fl(T)31669 13836 y Fp(\242)p Fv(\247a)p Fi(:)0 17242 y Fp(T)-101 b(o)405 b(\257nd)g(the)f(v)-67 b(alue)404 b(of)g(\242)p Fv(\247)g Fp(that)h(maximizes)e(this,)i(w)-34 b(e)404 b(\257rst)h(solv)-34 b(e)21443 19993 y(max)19058 20859 y Fm(\242)p Fo(\247)405 b Fm(:)f Fn(k)p Fm(\242)p Fo(\247)p Fn(k)24202 21008 y Fe(F)24881 20859 y Fn(\267)p Fm(1)26286 19993 y Fv(a)26964 19492 y Fl(T)27697 19993 y Fp(\242)p Fv(\247a)p Fi(:)19116 b Fp(\(41\))0 23399 y(Notice)404 b(that,)h(using)f(the)h(Cauc)-34 b(h)g(y-Sc)g(h)g(w)g(arz) 407 b(inequalit)-34 b(y)-101 b(,)403 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)12302 26149 y Fv(a)12980 25649 y Fl(T)13713 26149 y Fp(\242)p Fv(\247a)337 b Fj(\267)f(k)p Fv(a)p Fj(k)19914 26331 y Fm(2)20440 26149 y Fj(k)p Fp(\242)p Fv(\247a)p Fj(k)24347 26331 y Fm(2)25209 26149 y Fj(\267)g(k)p Fv(a)p Fj(k)28378 26331 y Fm(2)28904 26149 y Fj(k)p Fp(\242)p Fv(\247)p Fj(k)32133 26337 y Fl(F)32914 26149 y Fj(k)p Fv(a)p Fj(k)34804 26331 y Fm(2)35666 26149 y Fj(\267)h(k)p Fv(a)p Fj(k)38836 25649 y Fm(2)38836 26449 y(2)39361 26149 y Fi(;)0 28900 y Fp(b)34 b(ecause)372 b(of)h(compatibilit)-34 b(y)372 b(of)h(the)g(F)-101 b(rob)34 b(enius)373 b(matrix)f(norm)g(and) i(the)e(euclidean)g(v)-34 b(ector)372 b(norm)h(and)g(b)34 b(ecause)0 30405 y Fj(k)p Fp(\242)p Fv(\247)p Fj(k)3229 30593 y Fl(F)4347 30405 y Fj(\267)337 b 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y(Suyk)g(ens,)559 b(J.)528 b(A.)g(K.)f(and)i(V)-101 b(andew)-34 b(alle,)559 b(J.)527 b(\(1999\).)907 b(Least)528 b(squares)g(supp)34 b(ort)529 b(v)-34 b(ector)528 b(mac)-34 b(hine)528 b(classi\257ers.)1212 68679 y Fh(Neur)-62 b(al)432 b(Pr)-62 b(o)g(c)g(essing)432 b(L)-62 b(etters)p Fp(,)402 b(9\(3\):293{300.)25394 72000 y(29)p eop %%Page: 30 30 30 29 bop 0 1218 a Fp(Tipping,)384 b(M.)378 b(E.)g(\(2000\).)496 b(The)379 b(relev)-67 b(ance)376 b(v)-34 b(ector)378 b(mac)-34 b(hine.)495 b(In)379 b(Solla,)k(S.,)g(Leen,)f(T.)d(K.,)j(and) d(Muller,)k(K.-R.,)1212 2723 y(editors,)404 b Fh(A)-62 b(dvanc)g(es)432 b(in)h(Neur)-62 b(al)433 b(Information)g(Pr)-62 b(o)g(c)g(essing)431 b(Systems)i(12)p Fp(.)403 b(Morgan)h(Kaufmann.)0 5225 y(V)-101 b(apnik,)404 b(V.)g(N.)g(\(1999\).)538 b Fh(The)433 b(Natur)-62 b(e)432 b(of)h(Statistic)-62 b(al)431 b(L)-62 b(e)g(arning)433 b(The)-62 b(ory)p Fp(.)536 b(Springer,)404 b(2nd)h(edition.)25394 72000 y(30)p eop %%Trailer end userdict /end-hook 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