Nonlinear Data Representation for Visual Learning
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Nonlinear Data Representation for Visual Learning Bernard Chalmond, St´ephane Girard. No 3550 Novembre 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.
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(47) Vuª iZdSYcSU^KHÅGrIuXrVKSUW]c±i!Zd\aSUSUL,^bc³uM5X..^uitVKcdIKS]ltS]\.M5X ^u~aZdXeSUM \aW>f ^Qbz\ai!g_Xa W]°SULN^KSU`3^K\aS]«ÕLNSUÅ ^zSUZ[Á ·!cSUQu^K ¬X_ < XxKz f pvSYzNchZdTh\axa ltvcd =¨S]2\.^GwÅ S]Á ^ fQ{MSUj c ¤ Iu/ \.0 cdM¥M!jb`KldM0cdcdSU\a^SYc# ¡\al SU^uchZ[Xa^uiM6# < xav=¨¹ #
(48) # > v
(49) # 9
(50) & F «x.¬ / & # # ^ZdIuSUcmcd`KSUldS]Z5.Å Á QZ[M5itIK^uSU®zVKMIuX.c'sM5M!^Xe`K`uW]SUM f Zd\KVKSUW f cdM!W] ¦¨\al[.Xe^uS]·5M f ^uM!VKltXeWT^KM!Zo¤¢\.ld§bc¤ IKSYitI .l[M!S]~.WYX MZ[M X(fGW]\ f ¤XeZ[¦ fXSUL,cdMM5Z0^uk cd¨S]Z\.^ugTXeSUM!WW Lf c±XaXa`K^G`KSU^KVK(^b¡\.\a2W f ZdSUIK^KMN(cd\eVK'KZdLIKM Xe^Kf S]X ¡\ZtXf cd\aVKKXLIKXeSU^KaS]IK¡\a¦ fW f SULN M! ^uMcdSU\acd^uIuXaXaWW]WXei^ \af LN^KM\as^|XaitW]SU§^KM0WUXeXeZdl[M!W]g l Z[\,ZdIuXeZr§zSU^ f \eS]^ f Mj «Õ}¯#z kg^©ZdXIKª±M1>¬r¦chfZdM0IuiSULN\.Xe^Z¥M!f^u¤cdchMSUM0\aiWU^uSUZ[L,XaS]\.W¢S]^Z¥iVu\eZdK\ld~.Z[\.MIK^KSYc#,M`uXeXajb`s SU^ M5cl5«\e 7Zd¤Iu M M!lf 0M¤¬³u\a^K¡\.l Ml¥f X'c[Xe}¤§.\aMM^ch\eM0¯#XeS]^ui!ltWUitM5XaI&XeldlS]ZdZokm\gal[fS]^u±VMi!Z[KS]M!`sl±l[XeL,XaWKS]SU©¢L ^KM\aLNSYcX `sZdi\.\(Vu^KldXaM!~.`K^zM`KZ2l[ªr\Cjb^u¤ S]XaLW]IKgbX SYcditZ[SUI Mc M0¤ coS]Z[ZdS]IGLXXeZdi!M5VKc±ldXe~.^GM±l[S fM5i!M5\aXa^uWchi!ZdVKl[l[Vu~aiM ZdSU\a^`KVuldch\.SU^KKWUM!L_ZdIK¢M©¢c[XeVKLNl[~a`KMWU@M ldM0Gi\.^uKcoZ[l[ld\aVsLiZdSUZd\aIK^(SYc±Iu`sXa\.crS]^zM²Z±\aiSU2M!~z^zSUZrM!¤ ch\.3W]VK¤ZdMSU\aIu^uXCc ~a¤ MIuZdM!\ ^f ZdM0IKXeM W
(51) XaldM±\al f M!l[M f Xa^ f X`uXal[XaLNMZdl[SU·5X Z[S]\.^\eZdIKM iVuld~.MSYcaSU~aM!^#'QT\sbW]SU§aM±¡\al Å Á QZdM5itIu^KSU®zVKM.b\aVKl \aSYXac5#`K`KZ[l[IKl[\a\.M VKX.`s.itIK\.INS]W]^zgi!\aZtHccd^u`scd
(52) M0SY.coXeZtn §Tc¨SU^KZSU^u¡\acdWUM5WUSXe\ Blt¤rOitcIKNSUZdM ^KIuLXeZrX\a`uQ VK8XelltL XeLNLN M MZ[P!Z[IKld3S\ SU#·5f " XeQ>crZdP;SUcd\aIu^Q%Xe$>WUX O3WjbM8¤¢SYc\.LNP&ld¤ §Iu ¤SUP(itM!I'NWUP-WgTQ*¤ SU)nM!IKW Pf M5c^(L,Xe+-Z[^IK"/. M XaSU`Ki^ VK`Kl[l[~a\CM¥jT, SUZ[L\,X l[^Z[M5Mri!\a\a^ul f ch¢ZdM!l[l[ZdVuSUIu^KiSU cZ XM!ajT`KZd`KM!l[^ \.f X.M itf I(ZdSYc \WUcdM5M!c[~.crM!altXeM5^KWM!X ltjbXeSYW>Tc Z[ Iu7Xe0^_Z[ IKMm`KSUl[^uSU^uXeiWUWUSUga`uKXeSUW^iVKZ[IKl[~aMM0¡c \aVu\alh^KZ[M.I_ cd M5^_iZdZ[SUIK\aM ^bZ[IKM!jbS]l `sf M5ldcdSUM5LNiZ[M!S]^z\./ Z[^0 XaKWZ[ldIKM0SYchcVKXeW]Z[^s/ c0 Xe¤ WUgTSUcdWUSYWcsSUMc & f M5c[il[S]3M f .
(53). . . . . . . . . . . . . . . . . . . . (!(
(54) 3254. 1. X[)Z . ;. 87 ( :9. 6. g c BeHi a-c`fe g. #¯MZSUcVuXc^Ka\altX^bf ¦¨SYVschXe\aWUZdW]l[gG\a`3SU^zMEZdl[x\ Xef VuVuc[icdMSUXa\.^ VKliWU\aXeVWUaf \.ldXeS]^ZdIuf LZdITS]Vu^%c±Xak^&©S]ªÄ^zZdVuSUc¥S Z[¤S]~.M!MNWUWXaX cdf IKXeS]`K\.Zd^&M f Vu chSU^K¸XaitISUu`3Þ\apbSU^zZK ^
(55) ,SUSUu^9ÞpK«XzSU¬c `uXa`Kld\a`Knol[M5\CijbZ[S]S]L\.^X Z[SYMc!f¹ Tg|S]Z[c`Kl[\enoM0iZdSU\a^&\.^&ZdIKM(`KldSU^ui!S]`uXaW¢XejTSYc f M³u^KM f Tg Z[IKMVu^KS Z,~aM0iZd\.l ( HIuSUc . 6 ( C
(56) & ( (
(57) & F ^|SUuÞpK« ie f ¬Z[IKMi!W]\.V f SYcr^K\(LN\al[MExXeVuc[cdSUXa^_Xa^ f Z[IzVscS]Z[c¥Xe`K`uld\CjbSULNXeZdSU\a^_Tg(`uld\anoM5iZ[S]\.^ \.cdLN^\TZd\eIKZdM I_`KiSU^uVui!ld~.S]`uM¥Xa¤ WXeS]ZdjTI_SYc`uSYXac l[XaS]^sLNXaMi!Zdi!M5VKl=ltX < ZdM.n\ ^(^GZdIuXeS]sZ pui!X.«¡schM.¬uT¤ZdIKMMyXa
(58) ldM Xa^s`KX `KZdVul[\Cl[XajTW]SUWULgX WUZ[M S]f \.^Zd\SUc5Xa¹ `K`Kl[\CjTSULX Z[MyTg(X ( 6 A ( )
(59) & ? >A@CBED (
(60) F ¤ S Z[I A ( ( < ( & F \.>^SYcmZ[IKX±M¡VKX ^ujbiSYc Z[S]\.(^ \e±^!GZ[ IKSUc( ³s#
(61) a2VK¯l[MaM!zZ'ZdVsIuc2M¥^K\.\el Z[f MM!ZdlIu\eXeZHZdGIK M: G SU( ^
(62) / r 0 SUcXaZ[`KIK`KMrl[\Cijb\TS]\.Ll f X SUZ[^uS]~.XeM!ZdWUMg\ei3\aZdl[IKl[MXM5cd
(63) `s \.`u^ ldf \acnoZdM5\,iZ[ZdS]\.IK^ M. . . . . . . . . d}. "! #%$.
(64) Observations. Modele ACP. Modele non-lineaire. « f SUa¬VKkml[l[M\eno.M0¹¥iZd«ÕX.SU\a¬ ^H\al[^(Xa^uZdcdIuWUXeMZd³uM fl[chZri!VKkl[©~aM5ª7c5 `K«¡WYsXe^u¬±MaQT SULVKWYX Z[S]\.^GVuchSU^Kk©ª2«i0¬QTSULVKWYX Z[S]\.^GVucdS]^K(}¯k©ª. }~} n× +++.
(65) ¶Z[IKZd\.M`s`K\.l[W]S]\.^uai!SYS]i!`sXeXeWW\aX l jbf SYM!c l ¶ \af m\TZdM5IKcM@^u
(66) \eNZ S]Xe^ WUW]\ / 0 ¤µ, Z[¹r\ ZdIuf SUMc¥Z[M!SYl[c¥L,XSU^uiXel[VuZdM±i!SUZdXaIKWM`3Z[\a\aSU^z`3Z5\a WU\a±^aSYi!XaZ[WsIK\.Ml f iM!\.l0^.Z[l[uXa\aldlg.sS]^sSU^|coZtXe^uSUui!MaTpuZo¤«i5\ ¬ ^uX_\ai!^_ldVu^Ki!M!SUSUXaaW'IT`K3ld\a\.lrKXaWUM!^ L_f_Ef ( SU¨chZ[Xa¤^zMNZrch`sZd\.SUWUS]Wm^zX Z[crZhZ[S]M!^ LN/ `b0 Z , Z[LN\_XCXag(`K`KIul[XC\C~ajbMS]L^KM5X S]Z[.M6ITs \.l Tg`KldX_\anochM5LNiZd\TSU\a\e^uZ[I´c\.i!^(VKl[Z[~aIKM,M`uX XejbltSYXec LNM( Z[ ld¨SU·!ZrM SUf c Z[Ti5IKXeg M,^&G ¡3VKM^ui\ Zd~aSU( \aM5
(67) #l[^ i!\a>AZdLNIK@CBYMM f G l[&M!z`Kg´¤¢l[M5\.itcdVKIKM!W \T^zf \.ZtX cd^KS]Zd\e^uSUZ\a_^ f XaM5¤¢^W]SU\.~aX VKM5jbWl±fSUcl[M53¤ XaMIKW]SYSUS]ch^sitZdI&XSYif `KM0\bl[®.i!M5Vsi!cdVKX M!l[Zdl[~aM,ldM5M0¡^uc\.ilXaM0c cchgT¤\a^zh M!ZdWUuIKWM0Xach«ÕcSYc5`s¹¥SUu\z\.ÞcdpKVbcd«ÕZ[S]i5ucd¬dS W]f¬MNM,Z[HIKZdIKMIuMSUZdc \.~ `sXef WU\.S]VK²W]M0\.ic:aVKSYG i!W]ZoXa g W \.`ulldfM0chM!M5l±ld~ XeX ZdZ[M5S]\.l¥^`u\ald\asnoZ[M5IKiM Z[S]^K\.M!^=SUa«IT3SUu\aÞl[pKIK« \Tf \ ¬hf¬chZdKl[\.VuliZ[Z[VKIKldSUMc5\a¤" M,_cd'IuHXeWUIKW Mrf cdMM!³uWUM5^uiM,ZdM Xaf ^ XeSUjT^ SYf c'M¤ j SUW]/ WK LVXe jTSU& LNL,S]·5M0MXaZdcdIKVKSYl[c2S]^KSU^ f ZdMIKjM ¹ «¡Oz¬ ( / & SU^uXeWUWUgaTZdIuMchLN\T\eZ[S]^K,¡VK^siZdSU\a^_`uXal[XaLNMZdM5lSUc \.bZ[XaS]^uM f Xac¡\aWUWU\ ¤rc!¹ . . . . . <. . ( . @
(68) . ( C & (
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(75) ' HIuM¥M,Xa¤^uXeXeWU^zgzZ ZdSYi!XeWs
(76) 0M. jb`Kl[Z[M5\c[ch3SU\aM^Z[\aIK MN/ iSUWUc\.cd\aM5KchZ[Z±Xe^KSU^KM5MS].f IzXa3c\al¥¡\a\eWUW] \ ¤rc5
(77) '¯SUM^ Z / V & . !$#$%. . # ( & % # . '(.
(78) . . . #. . !$#$%. . #. . d}. "! #%$.
(79) . o o o. o. o o o o o o o o oo o o o. a. Sb 1. 1. o o. o o. o o o. o. o o. 1. o. o. o. o o. o. (b). o o. o. o o. o o o. o. (c). o. o. o. o. o. o. o o o oo o o o o o. o. o. oo o. o o. a. o. o. o. o. o. o. o. o. o. o. o. o. (a) S b1. o. o. o. o o. o. o o. o. a. o o. oo o. o. Sb 1. 1. o o o. o. o. o o. a. 1. o. o. o oo o o o o o. (d). o. SUaVuldMpb¹ «ÕX.¬XxXeVscdcdSUXa^i!W]\.V f «¡K¦¨i¦ f ¬}\a^8xXeVuc[cdSUXa^i!W]\.V f . }~} n× +++. o. o. o o. o.
(80) 0{ /0. ¢©¢\.^uchM0®zVKM!^zZdWUgaz¡\.lrM!~.M!l[g[ IuXacZ[\N~aM!l[S]¡g SU^KM5®zVuXaW]S]ZdSUM5c5¹ . . / & . . # ( & % #. . . .
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(95) i(!$# l[% M! .FGFldF M0)cd
(96) cd! SU# \a% ^ ( cdcd`KMZWUS]^u M G >\.@!|9;# :=% < ¯# GMZ& G G G . (( G@?B3MA C Z[G@IK?BM_DFchE \.+ lh=Z[M H#f \ cdMf Z5\|&ZdHIuXeIKZ5MH2Õ¦¡;\.ZdlNI M5iXa\Tit\aI l f iSU\T^u\aX l Z[f MSU^uchX LNZ[,M\T\etZ[mIK¤SU^KM( ^KSYcM5M `3f M!ldZd¡\&\al[chLN\.lhM Z,f Zd\.IK^ M Z[IKMGi\.VK`KWUM5c G@G ! %! C%
(97) G!$#$! % % & J FGF F C z=QTVuitI X|cd`KWUS]^KMl[M!`Kl[M5cdM!^zZ[XeZdSU\a^%SULN`KWUS]M0c,Zd\&itIK\T\.cdM .f M!SUc[^KiM5\.l[^.XaZ[WÕS]z^TZ[VKIKS]SUZocg_it§TIK^K\a\eSYiZtMcSYSUc ^G®.ZdVuIuS M Z[M G fG S ! ² % i!cdVKM!W l[ZSUM5Xac5^ ±f H¤ IKS]M,WUW^T3VKM Lf 3SUc[M!il¥Vs\ecdmcdM §Tf ^K\aS]^GZ[cªZd`KVK`s^KM5M0^c±f ZdS]IKjMN chLN\T\eZ[IKSU^KltX Z[MaK ^ . . X[ . 9ehlm^ Xc bdc`fehg % K.J b ¿ . À» ¡Z[HVu\6\^u
(98) #i`3Z[bM!S]Z[\.ldIK¡^\aM Äl[L QTSUZ[^u IK i!SYMcZd\.M!IK`bjbM Zd`KSUM0ldLNM0®.cdS]Vs·0cdX X SU\aZdZ[SU^|S]\a\.^ ^« e¬'¤¢cd
(99) MIK\ ;Is¤rXCcXD,~aM{S]LNZ[ \ `K WU^KSUM5\acZdMZ[IuZ[X ITIuZ gTX `sZ M53/ ld`uM! WUWU\aXaV^u^K .M5& ccSUZd^Kc,\,\aZdX_Z[M IKfM³uM ^KL ITS]ZdgzM(`3¹ M!`Kl[SU`KM5iWYXeM5^K¤ MSUcd\.M(lhZ[i!IK\a\a^u.ch\aZ[Xa^u^zXaZ W I" . /. . &. . . &. . . . L N . / 0 , F F 1 D@OH y5P'{ & 1 . N D
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