Séparation de sources. Principes et algorithmes - Gretsi

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Algebraic tools Statistical tools Criteria Spatial whitening Tensors Spatial whitening (2) Standardization via PCA Definition PCA is based on second or<strong>de</strong>r statistics Observed random variable x of dimension K. Then ∃(U, z): x = Uz, U unitary where Principal Components zi are uncorrelated ith column ui of U is called ith PC Loading vector Two possible calculations: EVD of Covariance Rx: Rx = UΣ 2 U H Sample estimate by SVD: X = UΣV H P.Comon & C.Jutten, Peyresq July 2011 BSS II.Tools and optimization criteria 42
Algebraic tools Statistical tools Criteria Spatial whitening Tensors Spatial whitening (3) Summary Find a linear transform L such that vector ˜x <strong>de</strong>f = Lx has unit covariance. Many possibilities, including: Remarks PCA yields ˜x = Σ −1 U H x Cholesky Rx = L L H yields ˜x = L −1 x Infinitely many possibilities: L is as good as L Q, for any unitary Q. If Rx not invertible, then L not invertible (ill-posed). One may use pseudo-inverse of Σ in PCA to compute L, or regularize Rx. P.Comon & C.Jutten, Peyresq July 2011 BSS II.Tools and optimization criteria 43
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Principle Algebraic tools Statistic
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Introduction Colored Nonstationary
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Lemma Problem Algorithms Biblio SCA
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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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