Séparation de sources. Principes et algorithmes - Gretsi

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Problème ACI Indépendance Indépendance de variables et vecteurs aléatoires Factorisation des lois ou densités de probabilité (ddp) Deux variables aléatoires 2 variables aléatoires S1 et S2 sont indépendantes ssi pS1S2 (u1, u2) = pS1 (u1)pS2 (u2) Vecteurs aléatoires Soit S un vecteur aléatoire à P composantes S1, . . . SP Les composantes sont mutuellement indépendantes ssi pS1S2...SP (u1, u2, . . . uP) = pS1 (u1)pS2 (u2) . . . pSP (uP) Commentaires Critère délicat à mettre en oeuvre Egalité entre deux fonctions multivariables P.Comon & C.Jutten, Peyresq July 2011 BSS I.Le problème et les principes de résolution 24
Problème ACI Indépendance Première fonction caractéristique Fonctions caractéristiques C’est la T. Fourier inverse de la ddp de S Conséquence : même information que dans la ddp Φ(ν) = E[exp(jν T S)] = . . . pS(u) exp(jν T S)du Seconde fonction caractéristique (SFC) C’est le log de la première fonction caractéristique Conséquence : même information que dans la ddp Ψ(ν) = log Φ(ν) Propriétés de la SFC Ψ(ν) = 0 pour ν = 0 Elle est continue et dérivable en ν = 0 P.Comon & C.Jutten, Peyresq July 2011 BSS I.Le problème et les principes de résolution 25
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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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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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