Mathematics in Independent Component Analysis

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104 Chapter 5. IEICE TF E87-A(9):2355-2363, 2004 IEICE TRANS. FUNDAMENTALS, VOL.E87–A, NO.9 SEPTEMBER 2004 PAPER Special Section on Nonlinear Theory and its Applications Quadratic independent component analysis SUMMARY The transformation of a data set using a second-order polynomial mapping to find statistically independent components is considered (quadratic independent component analysis or ICA). Based on overdetermined linear ICA, an algorithm together with separability conditions are given via linearization reduction. The linearization is achieved using a higher dimensional embedding defined by the linear parametrization of the monomials, which can also be applied for higher-order polynomials. The paper finishes with simulations for artificial data and natural images. key words: nonlinear independent component analysis, quadratic forms, nonlinear blind source separation, overdetermined blind source separation, natural images 1. Introduction The task of transforming a random vector into an independent one is called independent component analysis (ICA). ICA has been well-studied in the case of linear transformations [3, 12]. Nonlinear demixing into independent components is an important extension of linear ICA and we still do not have sufficient knowledge of this problem. It is easy to see that without any restrictions the class of ICA solutions is too large to be of any practical use [13]. Hence in nonlinear ICA, special nonlinearities are usually considered. In this paper, we treat polynomial nonlinearities, especially second-order monomials or quadratic forms. These represent a relatively simple class of nonlinearities, which can be investigated in detail. Several studies have employed quadratic forms as a generative process of data. Abed-Meraim et al. [1] suggested analyzing mixtures by second-order polynomials using a linearization in a similar way as we introduce in section 3.2, but for the mixtures, which in general destroys the assumption of independence. Leshem [15] proposed a whitening scheme based on quadratic forms in order to enhance linear separation of time-signals in algorithms such as SOBI. Similar quadratic mixing models are also considered in [8] and [10]. These are Manuscript received December 21, 2003. Manuscript revised April 2, 2004. Final manuscript received May 21, 2004. † The authors are with the Lab. for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, Wako-shi, Saitama 351-0198 Japan. ∗ On leave from the Institute of Biophysics, University of Regensburg, 93040 Regensburg, Germany. a) E-mail: fabian@theis.name b) E-mail: wakakoh@brain.riken.jp Fabian J. THEIS †∗a) and Wakako NAKAMURA †b) , Nonmembers studies in which the mixing model is assumed to be quadratic in contrast to the quadratic unmixing model used in this paper. For demixing into independent components by quadratic forms, Bartsch and Obermayer [2] suggested applying linear ICA to second-order terms of data. Hashimoto [9] suggested an algorithm based on minimization of Kullback-Leibler divergence. However, in these studies, the generative model of data was not defined and the interpretation of signals obtained by the separation was not given clearly; the focus was on the application to natural images. In this paper, we examine this quadratic demixing model. We define both generative model and demixing process of data explicitly to assume a one-to-one correspondence of the independent components with data. Using the analysis of overdetermined linear ICA, we discuss identifiability of this quadratic demixing model. We confirm that the algorithm proposed by Bartsch and Obermayer [2] can estimate the mixing process and retrieve the independent components correctly by simulation with artificial data. We also apply the quadratic demixing to natural image data. The paper is organized as follows: in the next section results about overdetermined ICA that is ICA of more sensors than sources are recalled and extended. Section 3 then introduces the quadratic ICA model and provides a separability result and an algorithm. The algorithms are then applied for artificial and natural data sets in section 4. 2. Overdetermined ICA Before defining the polynomial model, we have to study indeterminacies and algorithms of overdetermined independent component analysis. Its goal lies in the transformation of a given random vector x to an independent one with lower dimension. Overdetermined ICA is usually applied to solve the overdetermined blind source separation (overdetermined BSS) problem, where x is known to be a mixture of a lower number of independent source signals s. Overdetermined ICA in the context of BSS is well-known and understood [5, 14], but the indeterminacies in terms of the unmixing matrix of overdetermined ICA problem have to the knowledge of the authors not yet been analyzed. 1
Chapter 5. IEICE TF E87-A(9):2355-2363, 2004 105 2 2.1 Model The overdetermined ICA model can be formulated as follows: Let x be a given m-dimensional random vector. An n × m-matrix W with m > n ≥ 2 is called overdetermined ICA of x if y = Wx (1) is independent. In order to distinguish between overdetermined and ordinary ICA, in the case m = n we call W a square ICA of x. Here W is not assumed to have full rank as usual. Theorem 2.1 shows that under reasonable assumptions this automatically holds. Often overdetermined ICA is used to solve the overdetermined BSS problem given by x = As (2) where s is an n-dimensional independent random vector and A a m × n matrix with m > n ≥ 2. Note that A can be assumed to have full rank (rank A = n), otherwise the system could be reduced to the case n−1: If A = (a1, . . . , an) with columns ai, in the case of rank A < n we can without loss of generality assume that an = � n−1 i=1 λiai. Then x = As = n� j=1 n−1 ajsj = � (1 + λj)ajsj j=1 so source sn does not appear in the mixture in this case, thus the model can be reduced to the case n − 1. Overdetermined ICAs of x are usually considered solutions to this BSS problem. Often, overdetermined BSS is stated in the noisy case, x = As + ν (3) where ν is a decorrelated Gaussian ’noise’ random vector, independent of s. Without additional noise, the sources can be found by solving for example the square ICA problem, which is constructed from equation 1 after leaving away the last m − n observations given non degenerate projected mixture matrix. In the presence of noise usually projection by principal component analysis (PCA) is chosen in order to reduce this problem to the square case [14]. In the next section however, the indeterminacies of the noise-free models represented by equations 1 and 2 are studied because overdetermined ICA will only be needed later after reduction of the the bilinear model — and in this model we do not allow any noise. However, the overdetermined noisy model from equation 3 can easily be reduced to the noise-free model by including ν as additional sources: x = (A I) � s ν � . IEICE TRANS. FUNDAMENTALS, VOL.E87–A, NO.9 SEPTEMBER 2004 In this case n increases and we could possibly deal with underdetermined ICA (where extra care has to be taken with the now increased number of Gaussians in the sources). Uniqueness and separability results for this case are given in [7], which shows that the following theorems also hold in this noisy ICA model. 2.2 Indeterminacies The following theorem presents the indeterminacy of the unmixing matrix in the case of overdetermined mixtures, with the slight generalization that this unmixing matrix does not necessarily have to be of full rank. Later in this section we show that it is necessary to assume that the observed data set x is indeed a mixture. Theorem 2.1 (Indeterminacies of overdetermined ICA). Let m ≥ n ≥ 2. Let x = As as in the model of equation 2, and let the n × m matrix W be an overdetermined or square ICA of x such that at most one component of Wx is deterministic. Furthermore assume one of the following: i. s has at most one Gaussian component and the variances of s exist. ii. s has no Gaussian component. Then there exist a permutation matrix P and an invertible scaling matrix L with W = LP(A ⊤ A) −1 A ⊤ + C (4) where C is an n × m matrix with rows lying the kernel of A ⊤ that is with CA = 0. The converse also holds, i.e. if W fulfills equation 4 then Wx is independent. A less general form of this indeterminacy has been given by Joho et al. [14]. In the square case, the above theorem shows that it is not necessary to assume that mixing and especially demixing matrix have full rank if it is assumed that the transformation also has at most one deterministic component. Obviously, this assumption is not necessary if W is required to have full rank. Since rank A = n, the pseudo inverse (Moore- Penrose inverse) A + of A has the explicit form A + = (A ⊤ A) −1 A ⊤ . Note that A + A = I. So from equation 4 we get WA = LP. We remark as corollary to the above theorem that overdetermined ICA is separable, which means that the sources are uniquely (except for scaling and permutation) reconstructible, because for approximated sources y we get y = Wx = WAs = LPs. This of course is well-known because overdetermined BSS can be reduced to square BSS (m = n) by projection; yet the indeterminacies of the demixing matrix, which are simple permutation and scaling in the square case, are not so obvious for overdetermined BSS. Proof of theorem 2.1. Consider B := WA and y := Bs. Let b1, . . . , bn ∈ R n denote the (transposed) rows of B. Then
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