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Statistical performances of learning algorithm : Kernel Projection Machine and Kernel Principal Component Analysis

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(1)Statistical performances of learning algorithm : Kernel Projection Machine and Kernel Principal Component Analysis Laurent Zwald. To cite this version: Laurent Zwald. Statistical performances of learning algorithm : Kernel Projection Machine and Kernel Principal Component Analysis. Mathematics [math]. Université Paris Sud - Paris XI, 2005. English. �tel-00012011�. HAL Id: tel-00012011 https://tel.archives-ouvertes.fr/tel-00012011 Submitted on 22 Mar 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(23) š. £¤\¥-¦§$¥¦\¨ ª"×}áPÓº×}Ö«áPâãÓÑIâe¬$­ÉÖ2®Ô˜Ú¯¬ԘÓ׭ӘÜ. Ð6Ó7­ÉÖ°Ý^âãÝD­ˆÓºØ±ª"×}áPÓº×}Ö²³­}ÜPáPâK´ – †, ‡iƒG ‡iž/†Œ ‰ ˆ*¸5‚  ‡‰‚Rƒ-ˆ"¹ ¦ ‚  „*‚RŒ ‰ ‡K„&ž‡!¸·†Œ “ ˆIŸº¸5ˆI„*‚"„ ‘ „*†Œ»Sƒ‡iƒ •Iµ  h N{OHƒQ]}SÌ$OMVYQTN µµ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IµA‘ f HƒK$XYVWËvVWNI\PHƒVYK$L µ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IµA‘Iµ  m¹ K q VYXÏURKHƒOLGI\^Ì$KQÞê q VYXÏURKHƒOƒ¸  Ì ¹ ËvVY]IO[QTGRKHM\^OMQHƒL µ[µµ™µµµ[µµµ™µµµ[µ •IµA‘IµA‘  K$Ì$QTN]EQTHƒ]IKHFVWN{OMK$ZTHM\^XÉQTGRKHM\^OMQHƒL µ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IµA‘IµA• p \^VWNAêiHÍ\PËvK nFQTHMò\{N]_\^LMLSIËGOÍV»QNL µµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IµA• È KNKHM\^X  K$LSXYOML QTN  VYZPKN{ä†\^XWSK$LQ^ê È HÍ\PË p \ÞOHƒVYÌ$K$L µ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IµA•Iµ  QTNÌ$KNPOÍKHƒK$] g \^LMK µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ •IAµ ‘  K$Ì$KN{OMKHƒK$] g \^LMK µ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •Iµ½–  GIGXYVYÌ\^OMVYQTNEOMQ<k KHÍNK$X ¸ fºgF µ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •I½µ –}µ  t NÌ$KNPOMKHìK$] g \^LÍK µ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •I½µ –}Aµ ‘  K$Ì$KN{OMKHƒK$] g \^LMK µ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ ˜ g QTNÌ$XWSLMVYQTN6\PN]

(24) VYLMÌSLMLMVYQTN µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ   GIGRKN]IV ÊE•R ]I]IVYOMVYQTNI\^XGIHƒQQ^ê L µ µµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ Iµ  f HìQTQ^ê Lºê QHFLMK$Ì$OMVYQTN •Imµ ‘µ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ IAµ ‘ f HìQTQ^êlê QTH¡LMK$Ì$OMVYQTN •IAµ • µ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ IAµ • f HìQTQ^ê Lºê QHFLMK$Ì$OMVYQTN •I—µ –µ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ ’  GIGRKN]IV Ê6¶ }j QÌ\^X  \^]IKË\^Ì ¹ KH g QTËAGX»K Ê VYOMVYK$L µ µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ •IAµ ™  GIGRKN]IV Ê6g}j QÌ\^XYV ¾$K$]  \^]IKË\^Ì ¹ KH  äPKHÍ\^ZPK$LQTN  XYX»VWGLMQ{V»]IL µ8µµµ[µµµ™µµµ[µ •IAµ “  GIGRKN]IV Ê

(25) }g QTNÌ$KN{OHM\ÞOMVYQTN h NK$èISI\^XYVYOMVYK$L µ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ Ž „¿ / *‚ “ ˆ·„À*‚  Š‚"„Iž/‚<ˆ"¹ Š ‡Š"‚„Iƒ ¸·†ž/‚,ƒ<‡K„ ¦ ‚  „*‚,Œ  ‰  ‡„Iž‡!¸I†Œ “ ˆIŸº¸5ˆI„*‚„" ‘ „*†Œ‰Á. © ¢. r. »Sƒ‡iƒ –}µ  h N{OHƒQ]}SÌ$OMVYQTN µ+µ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µA‘  VÏHìLMO HƒK$LSXYO µEµµ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µA• h ËGIHƒQ^äPK$]  K$LSXYO µ_µµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µ½– g QTNÌ$XWSLMVYQTN6\PN]

(26) VYLMÌSLMLMVYQTN µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µA˜  GIGRKN]IV ÊE• \ p V Ê K$]  OîHM\^OMK$ZjAQ^ê  K$ZTSXW\PHƒV¾\^OMVYQTN µÉµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µA  GIGRKN]IV Ê6¶ }¶ \^Ì7ò{ZTHƒQSIN] Q^ê  SINÌ$OMVYQTNI\^X  NI\ÞXEjLMVYL µµ[µµµ[µµ™µµµ[µµµ™µµµ[µ –}µAIµ  h N{OMK$ZTHÍ\^OMVYQTN§Q^ê ¶ \PNI\ÞÌ ¹E GI\^Ì$K¼\^XWSK$]  SINÌ$OMVYQTNL µ[µ[µµ™µµµ[µµµ™µµµ[µ }– µAI µA‘ t LMKQ^êO ¹ K  K$LMQPXYäPKN{O µ±µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ }– µAI µA• ¶ \^LMVYÌ  K$LSXYOML µ µµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ ÃOÄ׋ª"×}áPÓº×}Ö«á{ÔÅ^×}ÑÜPânÔlÓÆ²Ç­ÉÑĺânÓº×. Ž. ©©. ¦. ˜Iµ  ˜IµA‘. ˜IµA• ˜Iµ½–. ‚  „ ‚,Œ ‰€ R ˆ ËG‚Rž/ ‡‰ˆI„.  †ž I‡„*‚ £ † ’ ‚

(27) Ì — ˆ ˆ*Œ^¹eˆ ‰ †,  ‚  „ ‹ ‚RžLˆŠI„I‡‰ ‡ˆ·„ h N{OHƒQ]}SÌ$OMVYQTN µEµ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ p Q{OMVYä†\^OMVYQTNL¡ê QTH¡O ¹  K k KHMNK$X f HƒQ{åæK$Ì$OMVYQTN p \^Ì ¹ VWNK µ µµ[µµµ[µµ™µµµ[µµµ™µµµ[µ ˜IµA‘Iµ  m¹ K È \PSLMLMVW\PN h NPOSVYOMVYQTN  \  O\ÞOMVYLMOMVYÌ$VW\PN  L f KHƒLGIK$Ì$OÍV»ä{K µµµµ[µµµ™µµµ[µ ˜IµA‘IµA‘ Ê OMKNLMVYQTN§OÍQ\ È KNKHÍ\^X g XW\^LMLMV ë}Ì\^OMVYQTN  HM\PËvKnFQTHMò µ±µ™µµµ[µµµ™µµµ[µ ˜IµA‘IµA• j VWNIO ò nVYO ¹ k KHMNK$X f HìVÏNÌ$VWGI\^X g QTËAGIQTNKN{O  NI\^X jTLMVYL µµµ™µµµ[µµµ™µµµ[µ m¹  K k KHMNK$X f HƒQ{åæK$Ì$OMVYQTN p \^Ì ¹ VWNK  XYZPQTHƒVYO ¹ Ë µµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ Ê GIKHƒVWËvKN{OML µEµ[µµµ™µµµ[µµ[µµµ™µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ ˜I½µ –}µ   VWHƒLMO s SI\^XYVYO\^OMVYäPK rÊ \PËAGX»K$L µµµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ ˜I½µ –}Aµ ‘ s SI\PN{OMVYO\^OMVYäPK rÊ GRKHƒVWËvKN{OML µ«µµµ[µµµ™µµµ[µµµ[µµ™µµµ[µµµ™µµµ[µ. µ·¶ ޼  ” ‘ ‘ • /– ’ ’ ’‘ ’’ ’’ ™ ” ™  ™‘ ™‘ ™• ™/– ™ ™’ ™“ ¼ ~ “‘ “• “˜ ““ ““  ” ‘  ” ‘  ” •  ” – ÈÉʵ ~RÍ*Î  ” ™  ” “  ” “  ”   ‘  •  •  –.

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