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Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis

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HAL Id: inria-00070768

https://hal.inria.fr/inria-00070768

Submitted on 19 May 2006

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Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis

Fabrice Rossi, Brieuc Conan-Guez

To cite this version:

Fabrice Rossi, Brieuc Conan-Guez. Functional Multi-Layer Perceptron: a Nonlinear Tool for Func-

tional Data Analysis. [Research Report] RR-5228, INRIA. 2004, pp.35. �inria-00070768�

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ISSN 0249-6399 ISRN INRIA/RR--5228--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COG

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis

Fabrice Rossi — Brieuc Conan-Guez

N° 5228

Juin 2004

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Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

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rE…w.¶Ab·…††`}qt¬‚lxa.…1oqbN‡otº…,uIxžIb3wc†`}bEr3xmqxtIwc§bEžvb3§X¶nkT…7z±Žwr;oxtIw…§V¨´ª®|5oq^…oŽmqbEm …º¸wxyob,w\Ža¶Ab3}stvz†…v}…vaTb3oqbE}m7­±¶AbErE…Žmqb7§xw`bN…}z±tv}a.m+r3…vw¶Ab}b3†`}bEmqb3w\oqbN‡.z±tI}+xwmpo…vwrb¦o^…w`Âlm

otSwnŽ`acb3}xœr3…§¨´ª=|m¯² 0bNmp†`xoqboq^b}…v‡lxœr3…v§°r^…vw`uvbxw0o^`bxw`†Žlocmq†…Irb‡`xacb3wXmpxtvw"­±z±}qtIa

R n

ot…r3tvac†…Ir;o·mqŽ`¶mqbotvz…cz±Ž`wroqxtvw…v§mq†…IrbN¯ÃX¤#brE…wSmpoqx§§b VbEroqxžvbE§k …†`†}qt¬‚lxac…oqbr3tvw\oqxwnŽ`tvŽm

z±Žwr;oxtIwm3²

ƒ oxmžvbE}qk rtvacactvw xw " 0s6oqt¹…ImqmqŽ`acbKo^…1o m€oއlxbE‡ z±Ž`wXr;oqxtvwXm…v}qb´mqact\tvoq^®Ãsoq^…oxœm…1o

§bE…Im€or3tvw\oqxwnŽ`tvŽmE²Tƒz#¤#bctvw`§kSrtIwmqx‡lbE}rtIaT†X…vr;o,xw`†`Žlomq†…IrbEm·z±tI}7oq^`t\mpbTz±Ž`wroqxtvwmEîo^`b3x}rE…vmqb

xœmcrt1žIb3}bE‡&¶nk5rtI}qtI§§œ…}k I²0ƒ«w‡lb3bN‡£Ãr3tvw\oqxwnŽ`tvŽmz±Ž`wroqxtvwm ­{tv}cactv}b}b3uvާ…v}z±Ž`wroqxtvwm¯ºtvw…

r3tvac†…Ir;o¦mqŽ`¶mqbo

Z

tvz

R n

…}btI¶nž\xtvŽXmp§kbE§bEacb3w\om·tz

L (λ)

¤+^`bE}qb

λ

xmoq^`bT}bEmpoq}xroqxtvwKtzoq^`b ª=b3¶AbEmquvŽ`bKacbN…vmqŽ`}qb ot

Z

²h¨´tv}b3t1žvbE}¥…0rtIaT†X…vr;ompŽ`¶Xmpb3o

K

tz…5mq†…vr3bKtz,}b3uvާ…v}.z±Ž`wroqxtvwm

­ØrtIwmpxœ‡lbE}qbN‡S¤+xoq^&o^`b.Ž`w`xz±tv}a w`tI}qa¥¯¦xœm…KrtIac†…vrompŽ`¶Xmpb3otz

L (λ)

² ]^`xœm,acbE…wXm¦oq^…oº…wnk r3tvw\oqxwnŽ`tvŽXm®z±Ž`wXr;oqxtvwºz±}tva

K

oqt

R

r3…vwº¶Ab+…†`†}qt¬‚lxac…oqbN‡,¶nk…sz±Ž`wXr;oqxtvwX…§`¨´ª=|…Im§tvwu¦…Im

L 1 (λ)

rE…w …§œmptT¶Ab…†`†`}t¬‚lxa.…oqbE‡Œ­ßo^`xmr3…w ¶Ab‡ltvwb·o^…wÂnm#ot

S U n

Ÿ! 9¬¢ß¯²

(W‚nob3wmqxtIw¹tvz·†`}tv†AtImqbE‡rtv}tv§§œ…}xbNm,oqtaާyox†`§b z±Ž`wroqxtvw…v§xw`†Žlomcxœm.m€o}…vxuI^\opz±tv}¤…}‡£²&ƒ«w

z{…Ir;oNÃvrtI}qtI§§œ…}xbNm=…v}qb°¶…vmqbE‡tvwº…v†`†`}t¬‚lxa.…1oxtIwtzX§xw`bE…v}z±tv}acmtIw

X

o^`bxw†`Žlo2mp†X…vrb#tzAb‚noqbEw‡lbN‡

wb3Ž`}tvwmE²W³ ^`bEw

X = L p 1 (µ 1 ) × . . . × L p r (µ r )

ÃX…v†`†`}t¬‚nxa.…1oxtIwtzCb3§b3acbEwIom+tz

X

xœm+tv¶`o…xw`bN‡

o^…wÂnmsoqt…v†`†`}t¬‚lxa.…1oxtIwmtzWbE§bEaTbEw\omstvz

(L p i (µ i ))

ÃA¶XbNr3…ŽXmpb…c§xw`bE…v}sz±tv}a!tvw

X

xœm….§xwbE…}

r3tvaº¶`xwX…1oqxtvwtz=§xw`bE…v}°z±tI}qa.m#tvw

L p i (µ i )

­ßo^`xœm#z{…vr;o+¤…vm#ŽmqbE‡oqtc‡lb¸Xw`b¦oq^`b¦z±Ž`wroqxtvw…v§£w`b3Ž}qtIwX¯;²

<= <®+° / 5& +,C 3

* $

·mb3‚l†`§…vxwbE‡Kxw´o^`bºxw\oq}tl‡lŽr;oxtIw®ÃVtvŽ`}¦uvt\…§xmoqtb‚l†`§œ…xwS…¥o…}uvb3ož1…}x…v¶`§b

t ∈ R o

o^…wÂnmoqt z±Žwr;oxtIw…§tv¶XmpbE}qž1…1oxtIwm

(g 1 , . . . , g s )

² A…Impxœr3…v§§kvÃW¤#b…ImqmqŽ`acb¥o^…1ocoq^b3}bxœmT…´z±Ž`wroqxtvw…v§#}b3§œ…1©

oxtIwmq^`x†0mqŽr^oq^…o

t ' F (g 1 , . . . , g s )

…vw‡¤°bco}qk´oqtKactl‡lb3§

F

o^…w`Âlm¦ot´…z±Ž`wr;oxtIw…§°¨´ª®|C² ]^X…w`Âlm#oqt.Žw`xžIb3}mq…v§£…†`†`}t¬‚lxa.…oqxtvw}bEmqŽ`§omuIxžIb3wxw o^`b,†`}b3žnxtIŽmmpbNr;oxtIw®Ã`¤#b,Â\wt1¤ oq^X…1o…wnk

}b3uIŽ`§œ…}

F

rE…w´¶Abc…†`†`}t¬‚lxa.…oqbE‡¶nk´….z±Žwr;oxtIw…§C¨´ª®|C²!sbEžvbE}po^`b3§bEmm3ÃV…wSxac†Atv}qo…w\o7†`}tv¶`§b3a }b3a.…vxwm

F

xœm+tv¶nžnxtvŽmq§kcŽ`wÂ\wt1¤+w…vw‡ …Tr3tv}}qbNr;o+…†`†}qt¬‚lxac…oqxtvw…Im#oqt.¶Ab,rtvwXm€o}qŽroqbN‡¥o^…w`Âlm ot.…º§xaTxoqbN‡ wnŽ`a¶Ab3}+tvzb‚`…vaT†§bNmtzoq^`xœm+a.…†`†xw`uX²

( * "+ ! * ) *

+ šv—5À ™'+Á™l—N™

ª=boŽm#w`t1¤ ‡lbNmqr3}qx¶Xb·oq^b·†`}tv¶X…¶`x§xœmpoqxœr+z±}…acb3¤#tv}ºtvz=tvŽ}°†`}tv¶§bEa²W§§V}…w‡ltIa IŽX…w\oqxoqxbEm#¤+x§§

¶Ab‡lb¸wbE‡ºtvwº…·uvxžvbEw†`}tv¶…v¶`x§xo€k,mp†…Irb

(Ω, A, P )

²"`tI}=o^`b+mq…vÂvbtvzXmqxac†`§xr3xyo€kIÃ1¤°brtIwmqx‡lbE}tIw`§k o^`b¥r3…Impbctvz…vwŽw`x&\Ž`bcz±Žwr;oxtIw…§2xw`†`Ž`oE²¨KtI}qbc†`}bEr3xmqb3§kvä°bca.…ÂIbToq^`b.z±tI§§t1¤+xw`u ^\kn†Ato^`bEmqxm

­

H b

¯

(13)

d [1 K

Z

xœm+…Tr3tvac†…Ir;osmp޶mpb3ostz

R u

Šl²

(G i , T i ) i∈ N

xœm+…wxزxز‡£²2mqb\Ž`bEwrb¦tz=}…wX‡ltva b3§b3acbEwIom¤+xyo^ž1…v§Ž`bNm#xw

C(Z, R ) × R o

­{xزbv²Ã

bE…Ir^

G i

xœm¦…¥acbE…ImpŽ`}…¶§b,z±Žwr;oxtIw´z±}tva

oqt

C(Z, R )

r3tvwmqx‡`b3}bE‡ ¤+xoq^Sxom A#tv}b3§mpxuva.…

…§uvbE¶`}…T…wX‡bN…vr^

T i

xm+…T}…w‡`tva žIbEr;otv}xw

R o

Ã`…w‡o^`bmpb \Ž`b3wr3b7xmxG²xG²‡£²¯

kn†Xtvoq^`bNmpxœmCtvwºoq^`b+tI¶mqb3}žvbE‡z±Ž`wroqxtvwmW…v}qb \Ž`xoqb+‡lxAbE}qbEwIoWz±}tva oq^`t\mpbtzVrtI}qtI§§œ…}xbNm +…vw‡TŠ =tIw

o^`btIw`b^X…w‡

H b

…}bmpoq}tvw`uIb3}oq^…vwSrtI}qtI§§œ…}xbNm^nk\†Ato^`bEmqxœm·…Imso^`b3kKrtIwmqx‡lbE}tIw`§kKr3tvw\oqxwnŽ`tvŽm

z±Žwr;oxtIwm2‡lb3¸w`bE‡ctIw.…,rtvac†…Ir;omqboNÃvtvwTo^`bsto^`b3}^…w‡ºo^`b3kc…}b¤°bN…ÂvbE}W…vmWtI¶mpbE}qžIbE‡z±Ž`wroqxtvwm

‡`tTwtos¶Ab3§tvw`uToqt.…crtIac†…vro+mqŽ`¶mqbostvz

C(Z, R )

²

™l˜1™' ¼ —E˜ ؚ À Á ¼ +

³b¦oq}kcoqtTactn‡`b3§Voq^b·}b3§œ…1oxtIwmp^x†¶Xb3o€¤°bEb3w

G i

…vw‡

T i

o^…w`Âlm°oqtc…Tmp†AbEr3x…v§AÂnxw‡tz=†…v}…vacboq}xœr actl‡lbE§2­{…c†…v}…vaTb3oq}xrz±Ž`wroqxtvw…v§=¨´ª=|°¯°oq^…os^…vm#o^`b7z±tv§§t1¤+xw`uºz±tv}a

H(w, g) = U

w 0 , Z

F 1 (w 1 , x) g(x)dµ(x), . . . , Z

F k (w k , x) g(x)dµ(x)

,

­G¡¯

¤+^b3}b

w = (w 0 , w 1 , . . . , w k ) ∈ W = W 0 × W 1 × . . . × W k

Ã\oq^b

F l

…}b†…v}…vacboq}xœraTtl‡lbE§m#…vm°xw

†X…}…acbo}qxœrwb3Ž`}tvwmEÃ

U

xmW…¦}b3uIŽ`§…v}z±Ž`wr;oxtIw.…w‡

µ

…¦¸wxyob+†Xt\mpxoqxžvbA#tv}b3§nacbE…ImpŽ`}b7­{‡lb3¸w`bE‡TtIw

Z

¯;²W]^`xœm2†X…}…acbo}qxœr#z±tv}a xœm$IŽxyobmpxacx§œ…}oqt,o^`btIw`bs†`}tv†AtImqbE‡cxw.o^`brtIw\oqb‚notvzin§xr3bƒ«wnžvbE}mqb

„sb3uv}bEmmqxtIwº¶nkŸ 5%1¢G²e·Ž`}2a.…vxw.actoxž1…oqxtvwc^`b3}b+xœmWoqtŽmpbs…uvb3wb3}…§nz±tv}a o^…1o#rt1žvbE}mz±Žwr;oxtIw…§

aºŽ`§oqx©G§œ…¬kvbE}†Ab3}rb3†`oq}tvwm¤+xyo^`tvŽloa.…Ânxw`uot\tºaŽr^^\kn†Ato^`bEmqxm°tvw¥oq^`bEx}+…}r^`xoqbNr;oqŽ}qb­{wnŽ`a¶Ab3}

tvz§œ…¬kvbE}mEÃl…vr;oxž1…oqxtvwz±Ž`wr;oxtIwmEÃ`§xw`bN…}oqbE}qa.mEÃlbor² ¯;² "`tI}+xwXm€o…wr3bvÃ`xz

U

xœm+‡lb¸wbE‡ …vm#z±tI§§t1¤sm

U (w 0 , o 1 , . . . , o k ) = X k l=1

a l T (b l + o l ),

­{‹I¯

¤+xoq^

w 0 = (a 1 , b 1 , . . . , a k , b k )

Ãnoq^b3w

H(w, g)

xœm#b‚`…vroq§kcoq^`b7tvŽ`oq†`Žlostvz…z±Ž`wr;oxtIw…§£tIw`b¦^`x‡‡lb3w

§œ…¬kvbE},†XbE}r3b3†lo}qtIw®ÃC…vmuIxžIb3w&¶nkŒb \Ž…1oxtIw0‘²Kmº…´mpxœ‡lb¥b AbNr;oNä#br3t1žvb3}…vwnkŒactl‡lb3§oq^…oŽmqbEm

xw\oqbEuv}…§œmotToq}…wmpz±tv}a …wxw`†`Žlo+z±Ž`wXr;oqxtvw xw\oqt¥…T}qbN…§VwnŽ`a¶Ab3}N²

intIacb,}qbNm€o}qxœr;oxtIwm…}b·w`bEbE‡lbN‡tvw

F l

z±Ž`wroqxtvwm…w‡tvw

U

­±^nkn†Xtvoq^`bNmpxœm

H c

¯

v²#z±tv}

0 ≤ l ≤ k

Ã

W l

xm+…cr3tvac†…vrosmpŽ`¶Xmpb3ostz

R v l

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1 ≤ l ≤ k

Ã

F l

xm+…ºz±Ž`wXr;oqxtvwz±}tva

W l × Z

ot

R

mqŽr^oq^…o

­{…\¯¥z±tI}+bE…Ir^

x ∈ Z

Ã

F l (., x)

xœm+rtIwIoxwnŽ`tIŽm

­±¶A¯¥z±tI}+bE…Ir^

w l ∈ W l

Ã

F l (w l , .)

xmacbE…ImpŽ}…v¶`§b

­{rN¯

F l

xm‡ltvacxw…1obE‡ tvw

W l

Ã#xG²bv²Ã°o^`b3}b xm…&aTbN…vmqŽ`}…¶`§bz±Žwr;oxtIw

d l ∈ L p (µ)

­±¤+xoq^

p ≥ 1

¯mqŽr^o^…1o+z±tI}z±tv}s…v§§

w ∈ W l

…vw‡

x ∈ Z

Ã

|F l (w, x)| ≤ d l (x)

²

’`²

U

xm+…vwŽ`wxyz±tI}qac§krtIw\oqxw\ŽtvŽm°z±Ž`wroqxtvwz±}tva

W 0 × R k

ot

R o

(14)

:1 CH

‘²

U

xm¶AtvŽ`wX‡lbE‡

kn†Xtvoq^`bNmpxœm

H c

…}b IŽxyob,w…1oŽ`}…v§®…w‡…}b¦z±Ž`§¸§§bE‡xw †`}…Ir;oxrE…§®mqbopoxwuIm

d°tvac†…Irxo€k¥tzoq^`b,†…v}…vacboqbE}+mp†…Irb7xœms…cr§œ…vmmpxœr3…v§A^nkn†Atoq^bEmqxmxwKrtvwXmpxœm€ob3wr3k¥}bEmqŽ`§yom3²

1mqbz±Ž`§Cr^`tIxr3bEmz±tI}

F l

…v}qbw\ŽaTbE}qxœr3…v§=¨´ª=| …w‡ ¶…Impxœmsb3‚l†…wmqxtvwmWz±tv}oq^`bz±tv}aTbE}EÃArtIwIoxy©

wnŽ`xyo€k.xœma.…w‡…1oqtI}qkTxw†`}…vroqxœrb¦…vm#tv†loxa.…v§A†X…}…acbob3}m…}btI¶lo…vxwbE‡.o^…wÂnm°otºuI}…I‡lxbEw\o

¶…vmqbE‡…§uvtI}qxoq^acm,­Ø…w‡o^`b3}bz±tI}qb

F l

xœm+xwuIb3w`bE}…v§£‡lxVb3}b3w\ox…v¶`§b7¤+xoq^ }bEmq†XbNr;o+oqt

w l

¯lz±tv}

oq^`b,§œ…1oqoqbE}EÃr3tvw\oqxwnŽ`xo€k.xm+tI¶nž\xtvŽXm#…Im

F l

xœm§xw`bN…}+¤+xoq^ }qbNmp†AbErooqt

w l

²

m¦m€o…1obE‡K¶Abz±tI}qbIÃX¤+^`bEw

F l

xmtv¶lo…xw`bE‡Koq^…vw`Âlm+oqt….wnŽ`acb3}xrE…§¨´ª®|CÃVxyo¦xm¦…¥r3tvw\oqxwnŽ`tvŽm z±Ž`wroqxtvw®²m

W l

…wX‡

Z

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F l

xœm,tv¶`o…xw`bN‡So^…wÂnm,oqtK¶…Impxœm7b‚l†…vwmpxtvw=Å w…oqŽ`}…§W^nkn†Xtvoq^`bNmpxœm,xœm7oqt …vmmpŽaTb¥oq^…oº¶…Impxœm7z±Ž`wroqxtvwmº¶Ab3§tvw`uKot

L p (µ)

²S]^`b3w=ÃCr3tvac†…Irxo€kŒtvz

W l

xac†`§xbNm…u\…xw oq^…o+oq^`b‡`tvacxwX…1oqxtvw^nkn†Xtvoq^`bNmpxœmxm+z±Ž`§y¸X§§bE‡£²

U

r3tv}}qbNmp†Atvw‡m2oto^`b7w`tvwz±Ž`wr;oxtIw…§V†…v}potz…z±Ž`wXr;oqxtvwX…§£¨´ª®|CÃlxo+xm#xwuIb3w`bE}…v§Xw…oqŽ`}…§

oqt …vmmpŽaTboq^…o¦xo¦xœm·Ž`w`xz±tv}aT§kKrtIwIoxwnŽ`tIŽmE²ƒ«w‡lbEbE‡£ÃV†Atv†`ާ…v}¦…Ir;oqxž1…1oxtIwKz±Ž`wroqxtvwm¦mpŽXr^

…vm

tanh

…vw‡oq^`b§tIuvxœm€oxr¦z±Ž`wXr;oqxtvw´…v}qb,Žw`xyz±tI}qac§kr3tvw\oqxwnŽ`tvŽXm+…w‡ actv}b3t1žIb3}NÃ

W 0

xm·rtIaº©

†…vroEão^`b3}bz±tI}qbº¤+^`b3w

U

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oq^`b…ImqmqŽ`ac†loxtIwo^…1o

U

xœm¶XtIŽ`w‡lbN‡xœm+…§œmqtTwX…1oqŽ}…v§Ø²

& —5™'+ À Á ¼ +2™

Á š À

–5ؖ3— ¼ š ½

]^bT§bE…v}qwxw`u†`^…ImpbcxwŒwb3Ž`}…§Ww`b3o€¤°tI}q´…v†`†`§xrE…1oxtIwm7rtvwXmpxœm€om7xwŒ¸w‡`xw`uo^`b.¶XbNm€o†X…}…acbob3}m

z±tI}s…cuvxžvbEwo…vmqÂA²Wƒ«w tIŽ`}z±}…vacb3¤#tv}ÂAÃn¤#b,…vmmpŽ`acb7uvxžvbEwK…T‡`xmpo…vwrb

c

tIw

R o

…vw‡¤°b,…ImqmqbEmm#oq^`b

\Ž…v§xo€kTtz£oq^b·w`bEŽ`}…§actl‡lbE§A…ooq^bb3ž1…v§Ž…oqxtvw¥†Atvxw\o

G i

oq^…vw`ÂlmWot

c(T i , H(G i , w))

²W³b¦‡lb3¸w`b o^`b,uv§tv¶X…§Vb3}}qtI}ac…I‡lb7¶nk.oq^`b,†X…}…acbo}qxœractl‡lbE§

H

z±tI}s†…v}…vaTb3oqbE}m

w ∈ W

¶nk

λ(w) = E c(T 1 , H(G 1 , w))

,

­+9I¯

¤+^b3}b

E

acbN…wm·b‚l†AbEr;o…1oxtIw®²,ª®bE…v}qwxw`uxm¦xw´z{…vro¦…†…}…acb3oqb3}·bEmpoqxa.…1oqxtvwS†`}qtI¶`§b3a:xwŒ¤+^`xœr^

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(15)

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(16)

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c ij

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(x j , g(x j ) + j ) j∈{1,...,m}

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