Mathematics in Independent Component Analysis

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14 Chapter 1. Statistical machine learning of biomedical data JD implies minimizing the squared sum of the off-diagonal elements of Â⊤CiÂ, i.e. minimizing f( Â) := K� �Â⊤CiÂ − diag(Â⊤CiÂ)�2F i=1 with respect to the orthogonal matrix Â, where diag(C) produces a matrix, where all off-diagonal elements of C have been set to zero, and �C�2 F := tr(CC⊤ ) denotes the squared Frobenius norm. A global minimum A of f is called a joint diagonalizer of C. Such a joint diagonalizer exists if and only if all elements of C commute. Algorithms for performing joint diagonalization include, among others, gradient descent on f( Â), Jacobi-like iterative construction of A by Givens rotation in two coordinates (Cardoso and Souloumiac, 1995), an extension minimizing a logarithmic version of (1.5) (Pham, 2001), an alternating optimization scheme switching between column and diagonal optimization (Yeredor, 2002) or more recently a linear least-squares algorithm for diagonalization (Ziehe et al., 2003b), where the latter three algorithms can also search for non-orthogonal matrices A. Note that in practice minimization of the off-sums only yields an approximate joint diagonalizer—in the case of finite samples, the source condition matrices are estimates and hence they only approximately share the same eigenstructure and do not fully commutate, so f( Â) from equation (1.5) cannot be rendered precisely zero but only approximately. Source conditions In order to get a well-defined source separation model, assumptions to the sources such as stochastic independence have to be formulated. In practice, the conditions are preferably given in terms of roots of some cost function that can easily be estimated. Here, we summarize some of the source conditions used in the literature; they are defined by a criterion specifying the diagonality of a set of matrices C(.) := {C1(.), . . . , CK(.)}, which can be estimated from the data. We require only that Ci(Wx) = WCi(x)W ⊤ (1.6) for some matrix W. Note that using the substitution ¯ Ci(x) := Ci(x) + Ci(x) ⊤ , we can assume Ci(x) to be symmetric. The actual source model is then defined by requiring the sources to fulfill Ci(s) = 0 for all i = 1, . . . , K. In table 1.1, we review some commonly used source conditions for an m-dimensional centered random vector x or a multivariate random process x(t) respectively. Searching for sources s := Wx fulfilling the source model means finding matrices W such that Ci(Wx) is diagonal for all i. Depending on the algorithm, whitening by PCA is performed as preprocessing to allow for a reduced search on the orthogonal group W ∈ O(n). This is equivalent to setting all source second-order statistics to I, and then searching only for rotations. In the case of K = 1, the search can be performed by eigenvalue decomposition of C1(˜x) of the source condition of the whitened mixtures ˜x; this is equivalent to solving the generalized eigenvalue decomposition (GEVD) problem for the matrix pencil (E(xx ⊤ ), C1(˜x)). Usually using more than one condition matrix increases the robustness of the proposed algorithm, and in these cases the algorithm performs orthogonal JD of C := {Ci(˜x)} e.g. by a Jacobi-type algorithm (Cardoso and Souloumiac, 1995). (1.5)
Table 1.1: BSS algorithms based on joint diagonalization (centered sources are assumed) 1.3. Dependent component analysis 15 algorithm source model condition matrices optimization algorithm FOBI (Cardoso and Souloumiac, independent i.i.d. sources contracted quadricovariance matrix EVD after PCA 1990) with Eij = I (GEVD) JADE (Cardoso and Souloumiac, independent i.i.d. sources contracted quadricovariance matri- orthogonal JD after 1993) ces PCA eJADE (Moreau, 2001) independent i.i.d. sources arbitrary-order cumulant matrices orthogonal JD after PCA HessianICA (Theis, 2004a, Yeredor, independent i.i.d. sources multiple Hessians Hlog ˆx(x 2000) (i) ) or Hlog px(x (i) orthogonal JD after ) PCA AMUSE (Molgedey and Schuster, wide-sense stationary s(t) with di- single autocovariance matrix 1994, Tong et al., 1991) agonal autocovariances E(x(t + τ)x(t) ⊤ EVD after PCA ) (GEVD) SOBI (Belouchrani et al., 1997), wide-sense stationary s(t) with di- multiple autocovariance matrices orthogonal JD after TDSEP (Ziehe and Mueller, 1998) agonal autocovariances PCA mdAMUSE (Theis et al., 2004e) s(t1, . . . , tM ) with diagonal autoco- single multidimensional autocovari- EVD after PCA variancesance matrix (1.7) (GEVD) mdSOBI (Schießl et al., 2000, Theis s(t1, . . . , tM ) with diagonal autoco- multidimensional autocovariance orthogonal JD after et al., 2004e) variances matrices (1.7) PCA JADET D (Müller et al., 1999) independent s(t) with diagonal au- cumulant and autocovariance ma- orthogonal JD after tocovariancestrices PCA SONS (Choi and Cichocki, 2000) non-stationary s(t) with diagonal (auto-)covariance matrices of win- orthogonal JD after (auto-)covariances dowed signals PCA ACDC (Yeredor, 2002), LSDIAG independent or auto-decorrelated covariance matrices and cumu- non-orthogonal JD (Ziehe et al., 2003b) s(t) lant/autocovariance matrices block-Gaussian likelihood (Pham block-Gaussian non-stationary s(t) (auto-)covariance matrices of win- non-orthogonal JD and Cardoso, 2001) dowed signals TFS (Belouchrani and Amin, 1998) s(t) from Cohen’s time-frequency spatial time-frequency distribution orthogonal JD after distributions (Cohen, 1995) matrices PCA FRT-based BSS (Karako-Eilon non-stationary s(t) with diagonal autocovariance of FRT-transformed (non-)orthogonal et al., 2003) block-spectra windowed signal JD ACMA (van der Veen and Paulraj, s(t) is of constant modulus (CM) independent vectors in ker 1996) ˆ P of model-matrix ˆ generalized Schur P QZ-decomp. stBSS (Theis et al., 2005a) spatiotemporal sources S := s(r, t) any of the above conditions for both X and X⊤ non-orthogonal JD group BSS (Theis, 2005a) group-dependent sources s(t) any of the above conditions block orthogonal JD after PCA
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298 BIBLIOGRAPHY Barber, C., Dobkin
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