Boxlets: a Fast Convolution Algorithm for Signal ... - Yann LeCun


    

 !
 "#$
%&! ('*) +,
% - !#./102%
4356((.# /*7 89*% :8;< >=?
%&@ ABDCFEHGJILKNMPORQSGJT$BDEHURVXWKY Z\[jlkn]^m Z_kCFC`Z_oqabpsr_VXtvAuxwnBDCFydEHtzydGJIdpLc{}|~eBDfb[gKE5Bh[XUM Bh[X[WRKhi^ag[ ˜`€`™ LpHƒ†}{z‚ ‹š||›~y!„hŽ …yF…‡†}œ!ˆ&’rq‰ŠŽ {}~‹psŒLž\yLŽh’wyF{`ydŽ Ÿ6ps‘>’h’¡ps’h s¢“,Žh{zy`”Št£•y`ps{–L|~S–sh¤ pH€ †£u †`–s¤ |FLœL—L¥— ¦§©¨LªD«h¬l­\ª t’h£‚ ‹‹œsŒL®L†ƒy ’pLpL„_t^… t£y&™ ‹¯¥ {}œLœD±²™ |›œL|›ydœL{zt}†}tz’pL‹¯Œ!’h’!œL®†µ…„DpLpL†z’tµ‹œL |›’SœL™ ¤b¥ ps³x†£’™ †}„hyF¥´{}†}’NpHps†}’{z‹¯œ!Ÿ y`’|›|dœ!pspL®L…¶tz’hydt ‹¯™td†zŽ‹yFœLy`|›’¡œ!¥¤/ps…™ ®L³x’·„Dœ!†}{zps±¸‹¯†zyd†z~ps‹œL†z¥ƒ„h’t{}pLy5…q¥ƒpLyFpL|d°“L|›†}y&„h{}p!{yFpL|¹°†}|F†z‹¯ŸyFœ!’h’S‹štu Ž ±¸DœL‹š{^tv†}‹¯œL’{zt£†z†}ydpLS’¤»|›k>y!Žq~†}‹tb~hy±¸œL±¸{}y`¥¼pH†}œs„h±g{zy`’ht œL‹šœst£±yb‹½v’„†zt£yd†z{z‹¯¾y`tvyd†&t>‹tz’œL¥ƒpNy©tz…¯‹¯yd®!ŒL’yFps…,…bœLps±g{}ºy´„„pst£’„_†zps‹ˆd……¯psŸ†z‹œLº’„h‹‡†}‹’ y œLœs±>{Dp yF™h{©™ †}{}œNœÁ°DpL‹¯|¥´~hps‹yF†zŒ!‹’hy^®±¿p!{}yFtv†}®LyF‹œL{(’_±¸tŠy`pHœL†z±R„{z†zy´~yƒyF°tz‹†}®L{}’p!|¹pL†}…X‹¯œ!Ân’S‹¯†z¤ps~h’‹št5r\y&¥ƒ{zyFy`†z|›~hœHœDŒ u †zŸœ´‹¯yd|›…hœ!t’ŒLt£œL„…r„Dtv†}†‹¯psœ!’’†zps‹š…,ps…X’hydt ™„h{}yFpLy`…¡’h„ y›™ †v¶‹’6œL{}±¸“DydtFps¤ †z„h{}yÄyF°†z{pL|¹†}‹¯œ!’ps’6‹štp ™h™ …‹|dpsrh…y Ê Ë!Ì ªD«hÍnΊϊ­\ªhÐ£Í Ì ³x‹’’ ŒH™pL{zpH‹š†£ps†}’yF|F{}y’N™ {}{zydœ |›™ œ!yd®L{£’h†}‹¯‹¯†zy`‹tFœL¤g’qÑhŽ\yd|FpsœL†z’„hŒL{}œ!yl…¯„Dy›°†}‹¯†}œ!{}’p!|¹‹†zt(‹œLpL’(’N‹štS‹¯¥ p¶™ †vŸ œL™{z†}‹š|FpLps’!…L†y›†z°hœpsœL¥ …»™ r\…ydyL|dÒSpsk>„~htzyÄybœLD±²‹t£‹‡†}†pLtŠ’†}|›{}ylpLr\’y›tz†v…ps¶†zyF‹œLyd’’ …±¸šp&psydps{}tz®L¥´†z„hyFps{n{}…yn…œ!†}’h™ yFyLps¥ ¤>†£†z™ k>yd…š{zpH~h’N†zy&yÅ¿{}‹JÂnyd¤ yLtz‹‡¤X„h†}~ƒ±¸…¯†zy`‹†zpH’h~h†z®·„y{z…ÔzyÁpLhÆ»{z‹®!t£‹tb†}yFpL{ |›’œ!™ |›¥ psy †£™ †}‹yF„D¥´{}†z’Spsy`®L¤»y`³xpHՃ’ƒ†b‹†zpLt~h…¯†vyŸ ™ {z™ œyd‹¥ƒt£|d‹¯ps†zpL…‹…¯‹¯œLŸ~±¶~hy ‹¯yd¥´{z†vÂby&pLœ5®LpLyd{z‹¯t(y ¥´œL¥´ps±b®!ps†}yd’~htnŸ y´‹’6¶tzpLpÁ†z¥ƒ~ŸDt¶y&yƒ†}œLtzœ5{}‹ˆF‹¯|›®!y´œ!‹¯’_’‹’ÉpsŒ!…qœL†}…t ~hŒL™ yy©pL|F‹¯Ñh¥´yLœ!¤„hpLÀŠ®L{z‹ydyF±Rt{&|›y›œ!hÖ„hœL{}¥´|›tz‹yFy©ps’‹†z’/†z~h…ŸL‹š|Ft¤nœL{z{}ѝy`{zœLy`º!{t„™ ‹‹¯’{}œLydt£’_tn†}hps×Pt’_|›†zØœyLŽ\…p6œLp´®¶|›¥ÄœLØْ„Œ!…‡œ†}œL‹™ ™…„DyF…{‹š†z|FpH‹œLpH†z†}’P‹œL‹¯œ!’_œs’ t± œLy›Å¸Ân°„h†}†~{}yFp!œs{}|¹±»y>†z†}‹ØÚ~hœL’Éy^‹štXѝr_œL†zy`~h„h|Fynps{}‹¯„_’ydt£{„hyÄ¥^D†zœ!r_~h¥´ydy^{lps±¸œsyd‹’Sps± ¤†z™ „h‹‡k>°D{}yÄ~hyF…šyd†}t»tzœ5y&œs±_r\¥ƒ†zy^~hy›yF†}yn°~h‹†zœD¥´{hpLpst©|¹®!†}psy¶yd{}pLy&’‹š„^t(tz×ۄt£¥´ps…‹špL…¯tlŸ …¯Xp(’hÂn|›œLœL‹‡††}’_~Pp tv†™h{}ps™yd’t{z†™œ Ɲ™yd½v|¹{z„†‹špHt£†z†»†}œ€ ±¸ŽœL{y›†}™ |LœL¤¯…¤ Ÿ’hœL¥ƒ‹šps…št>œs±lDyF®!{zydy  Å¿|›œ!’t£†}ps’†t}ƛŽ†z~hy ¾_{}t£†ŠÞ ÑXp ‹¯±¸®!{„hpL{}|›y†z‹ÜœL’´‹……¯œL„±¡t£†z†z~h{pHy†}œLyd{}tX‹¯†}®!~h‹¯’ypL{}…yd‹tz¥´„h…‡ps†®Lt»y!œs¤X±\k>†z~h~yyŠº!r\„_œsps†£’†}œL†z‹¥ ˆdps…†zy›‹±àœL†>’ƒ‹¯¥´ps…ps®L®!œLy{}‹¯†z‹…¯~h„¥´t£td†z{¤gpHk>†zy`~htly>†}†}~hœ y™ r\…¯yFœÁ±à°D†b…¯yF|›†zœ!‹{zˆd’psyF†z{»‹œL‹’ t œs{}‹¯±X®!†z~!~†gy ‹¯¥´®L{}psyF®!ydyRh‹Ÿ…¯„pst£…®L†z{œLpH{}†}‹¯†zyd~ht¡¥6†}~hŽDy¶Ânr_‹¯œÁ†z~°D…y›™ †zœ!‹…¯ˆdŸps’h†z‹œ!œL¥ƒ’&‹œLpL±D…t>†z~hœLyb±X{zDy`yd|›®L„h{}{yFtzy ‹¯Œ!€ yRŽhpspL…’®Lœ!{z‹¯†zP ~h# ¥6Ž`Ân€d‹‡ †}-~ !™LœL…¤XŸk>’h~hœL¥ƒy(†z‹špsœ …š™ t œsD±R‹št h™ yF…pÁ®LŸ{}yF‹’hy ®  †zps~h’y´ r\œÁP°D…¯# yF†(r\€dœL „- ’!hLps {}¤n‹ydk>td¤µ~hy^³x’/r_œL†}†£~h†}‹œLtÄ¥ |FpL{ztz‹yµ®L~†z†~hy ‹št™ †}‹‡~h°Dy^yF…»t}†}psœ¥ƒ‹¯yÄ¥ p!™ tn„†}…tzœ yƒ™ ±¸{}„h‹¯’®!~|›†z†Š‹œLÂn’‹‡†}{}~hpsœL†z„h‹œ † ÞD¤ >h¤ ? @ BÐ A?DS4« D Ì ªDУ¬XªDÐ£Í ?Ì >@ «h%Í A EHÍ GJI1DSªD¨/ªDÍ Ë A;C#Ï I£S¨ DEDŠÏ Ì ­\ªhÐ£Í Ì ¨ ³‹¥±  ™ P „h‹…t¶tzy&p ±¸™ „œL’…|¹Ÿ†}’h‹¯œ!œL’ ¥ƒ* t(‹špsD…ydœL{z±q‹ŒHDpHyF†}®!‹¯{zŒ!ydydy tdŽK_ÂnŽ!‹¯~h†}‹št©|~ÉÅ K ps& {}y&€ ÆxˆduJyF†}{}~5œDyFydŒ!{zyF‹{}ŒHŸpHÂn†z‹~ŒLyyF{}|dy psrh’„Dr\† yŠpHÂn†Š{}†}‹¯~h†£†zy^yd’5|FœLpL{}tb’hyFp&{tntz„hœL¥ ± OQœsP ± ¤ k>pL|›~h†zy`ydt£{zy ‹ˆF‹y ¥  ™ P ¤l„h…šk>t£y(~hyd±¸„hŸ´’_|F|¹ps†z’ ƛr_Ž!y(ÂnD‹¯†z‹¯ž\~’h~hœ!‹¥ƒtb‹‹špLtR…_‹…¯‹štb„p^t£†z|›{pHœ!†}’ydtv†psps’†R†dŽD†z~tzœy †z|›œyd™’†nœLt}±gºÑX„‹¯ps®!{}„hyd{zt¶y D— œƒ¤lt£k>‹¥ ~hy(™ …|›‹‡±¸œHŸ!Œ!ŽDyF†z{0~N ps†nr_‹tdydŽ‹¯’†z®µ~hy p t}™ psps¥ƒ{z†zy(‹¯†zD‹œLyF{}’S‹¯ŽŒHpsr\†zœL‹„hŒLyd’_t¶hpsœL{}±XŸ´p^|›‹¯œL¥ ’_™ D„h‹‡†}…š‹¯tzœ!y’±¸t¶„’r_|¹yF†}†v‹¯Âbœ!yd’yFtn’ pHpL†>Á†z½£~psy u †zt}Ÿœ´„ps¥ƒ™ Dyœƒ†}œµ…tzœœ^pÄ|dpHr\±¿pL†}yd‹¯|›|dœ!†zps’ƒœL„{ |dtzpsy€ ’´ ‹¯†n r\ÂnyŠßċ¯|›……¡œ!Ý¥Ä{}ßydrhDŽ‹„œL’h{¶|Fy`y©ƒ¥µ†zrœ!~hŸƒ{zy y©pL|›‹¯œ!±X¥Dtz‹¯œL’™ ¥ƒ®(„D†}yŠ†zps~hœL†zyd±q‹œL‹¯†z{>’~h’Pyƒ~hps‹š®Lt²…{}‹¯‹š®!yFt²’hyd†}hyd~h¡ŸNyŠŽ,ps{}tz…ydœ®Lp!œL†zt£{}œ!~_‹¯’´†zpH~h†ŠÂn¥6†}~~h¤&Ÿ^yµ³x†z’É‹¯~¥ y†z™ ~h{}„hyƒyd…š|Ftz{}„hy&yd{}|›±¸tz„„h‹ŒL{}’tzy|›‹¯†zŒ!ps‹y^…œL®L’pLœ!t©…¯{z®!‹¯œL†zœL~±R{}¥Ã‹‡pL†}Á~h±¸½£¥6œ!ps{u Ž |›yd’†&r_œÁ°D…y›†}t |dps’/r_y…†v„h¶t£yL„hyFŽ!yF¥ƒ†z’É~¥´y†zps~h†z{}yƒœ ‹ˆF™ t£yd†}…Äps‹¯’h’_‹¯y’ƒhps‹¯kX’_{psDNrh‹|d|›…ypHœ!†}’€ ydŒL¤gtlœLk>†}…~h„D~†}yyb‹¯œ!†}±¸‹¯œ!’N¥ƒ„hps{ly’…š{}NpLpst£†z†»†z‹~hœ&|FyµœLœs…r_±S„hœÁt£¥ƒ†}°Dps’…y›’_t»†(h‹¯’_ps|›œL{D´’‹|dŒ!|›pHœLœ!†}…’yb„DŒ!†z†}œL‹~hœL…y>„D’S†z¥ƒ¤(‹œLydÑh’ƒpLœ!tz{(Œ!„hyFyd{}{ydpLt£&|„~NtR†z‹†zº!¥µ~h„_y>{}psyd’{tzpH†z~h†}‹œLˆd‹¯œps…š tu †zD‹œœLyd’qt¶Žh’h|›œLœL’†bŒ!†œLps…“!„Dy†z‹‹œL’’ †zœƒpspL’|FœL’†}~h†zy ‹¥µr\y!œÁ°¤²…k>y›†~hy©|FœL±¸’ydŒLpsœ!†z„h…¯„D{}†}y©‹¯œ!tz’S‹ˆF¤»yŠk>Â>~hpLy(t¶r_ŒHpsœL{}†£†z‹ydœ!ƒ¥ ±¸{}…œL‹¯’¥ y ˆdtzÞ yFpsyF݆z’pL†}~’~h„´y©tŠ†z†}œÂn~h™ y&‹¯†z…t~/y›™ ±à†pyFy`ps†}5’~h„{}yd™ †ztz~h~h±¿p!y©œL|¹…šr\†zœ!œsœs{>†£±†}‹œLt €`¥  Ü Ž ¤ L>{}‹!¤»®L ~k>†bps~’‹y©¥´É…psœ!p†}™ ‹¯{z¥ƒœ yÁ™ ƹps¤ ®pH†}‹¯œ!’/‹’|FœL’ŒLœ!…¯„D†}‹¯œ!’ps…b’hyF„{}pL… k>pL|d~h|›y „h{’hpLyF|F°Ÿ´†Šœsp ±g™h†z™ ~h…‹šy&|FpH|F†}œL‹¯’œ!ŒL’ œ!tz…¯„D~h†}œH‹¯Âœ!’St>Žhp´ps’_|Fp!5t£y©Ÿ!y›Ân†n~h…pLyd{z{zy&®!y ºt„™ pLyF’!y`†}5‹¯ˆ`pH„ †}™ ‹¯tœ!’5ps{}~y(p!œLtnrD’h†œ´ps‹pL’hDyd¡Œ!yF¤ {t£y©y›ž,yd|›†œL’ †z~y  E·Ð Ì ¬»« Û=Ð A C¬ B Dq¨·¬ Ì Î «_Ï  Ì  I1D Ì Bl ª D Ì ­,ÍnÎÐ Ì B †znk>‹ŒL~h~hy y5yd’6‹šºt†z„|F~hpLœLy^’’†z‹¯tz‹¥´‹ˆdhpHpsyF†}®!{}‹¯y œ!yd¡’$‹tŠŽtv†}rh†}~hy‹y&’™ pst^‹¥{}ŸLD™ ¤¶yd„ht}³ |›…±»tz{}y †z‹r_~h{}y`yÄy ™ †z~h{}‹¯yd’ {}tzydyFtz‚ ’~hyd†}œL|¹ps†}…š†z ‹¯‹œ!œL‹’’ tt t£‹ÜyFt†ydps†zº’œ5„h ‹ˆFŒÁyF— pL{}…¯œydr_’Ž_y`†nps|›œL’†z¥ƒœ´6y {}‹¯±²„h™ ’DœLpsu’h{z…¯…yd†zŸ’h‹š|›®s†}„h~h†}…š~  Ž¡ŒLœL…Ó¤ Üs Ž ™h ™ ¤ —h€ Þ ! —!Üh€ ¤ Boxlets on binary image

Compact Dirac representation