Séparation de sources. Principes et algorithmes - Gretsi

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Problème ACI Indépendance Indépendance, fonction caractéristique et cumulants Indépendance et fonction caractéristique pS1S2...SP (u1, u2, . . . uP) = pS1 (u1)pS2 (u2) . . . pSP (uP) (8) Développement de la SFC Φ(ν) = Φ(ν1)Φ(ν2) . . . Φ(νP) (9) ln Φ(ν) = ln Φ(ν1) + . . . + ln Φ(νP) (10) Ψ(ν) = Ψ(ν1) + Ψ(ν2) + . . . + Ψ(νP) (11) le terme de gauche contient des termes croisés les termes de droites ne contiennent que des dérivées par rapport à une seule variable Problème indépendance ssi tous les cumulants croisés sont nuls Il y a une infinité de cumulants Peut-on s’arrêter à un ordre donné ? Comment le définir ? P.Comon & C.Jutten, Peyresq July 2011 BSS I.Le problème et les principes de résolution 28
Définition Problème ACI Indépendance Divergence de Kullback-Leibler KL(f g) = . . . f (u) log f (u) g(u) du KL(f g) mesure l’écart entre les deux distributions f et g c’est un scalaire positif qui s’annule ssi f = g (la démonstration est aisée en utilisant l’inégalité de Jensen) Divergence KL pour mesure l’indépendance KL(pYΠpYi ) = . . . pY(u) log pY(u) ΠpY i (ui ) du KL(pYΠpYi ) est un scalaire positif qui s’annule ssi les composantes du vecteur aléatoire Y sont mutuellement indépendantes On peut identifier cette mesure, KL(pYΠpYi ), à l’information mutuelle (IM) I (Y) définie en théorie de l’information P.Comon & C.Jutten, Peyresq July 2011 BSS I.Le problème et les principes de résolution 29
Page 1 and 2: Séparation de sources. Principes e
Page 3 and 4: Problème ACI Indépendance I. Le p
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Algebraic tools Statistical tools C
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Principle Algebraic tools Statistic
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Introduction Colored Nonstationary
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Intro Dim2 Orthogonal Deflation Dis
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Invertible Iterative SemiAlg ConclR
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Intro Inversibility Time Frequ VI.
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Intro Inversibility Time Frequ Nota
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Global filter Intro Inversibility T
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Interest Intro Inversibility Time F
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Kronecker notation Intro Inversibil
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⎛ ⎜ ⎝ Intro Inversibility Tim
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Source separation Intro Inversibili
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Fourier domain ? Intro Inversibilit
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Intro Inversibility Time Frequ Intr
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Identifiability ICA for NL Discussi
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Undeterminacy Identifiability ICA f
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Comments Identifiability ICA for NL
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Theorem Identifiability ICA for NL
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Identifiability Identifiability ICA
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Problem Algorithms Biblio SCA Conte
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Problem Algorithms Biblio SCA nCF I
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Problem Algorithms Biblio SCA Dim.2
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Lemma Problem Algorithms Biblio SCA
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Example [CR06] Problem Algorithms B
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Problem Algorithms Biblio SCA Spars
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32 Introduction Titre sous-section
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Introduction NMF NTF Titre sous-sec
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Introduction NMF NTF NN ICA Algorit
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Introduction NMF NTF NN ICA Algorit
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Newton-like method Introduction NMF
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Noisy mixtures Introduction NMF NTF
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Introduction NMF NTF NonNegative Te
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Bibliography XI. Bibliography I II
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[BCMT10] J. Brachat, P. Comon, B. M
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[Com95] P. Comon. Supervised classi
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[F´88] L. Féty. Méthodes de trai
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[LISC11] R. Llinares, J. Igual, A.
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[PAB + 02] M. Plumbley, S.A. Abdall
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[TF07] T. Tanaka and S. Fiori. Leas
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Séparation de sources. Principes et algorithmes - Gretsi

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