Mathematics in Independent Component Analysis

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116 Chapter 6. LNCS 3195:726-733, 2004 SOBI based on multi-dimensional autocovariances 3 for τ �= 0. By assumption ¯ Rs(τ) is diagonal, so equation 3 is an eigenvalue decomposition of the symmetric matrix ¯ Rx(τ). If we furthermore assume that ¯ Rx(τ) or equivalently ¯ Rs(τ) has n different eigenvalues, then the above decomposition i.e. A is uniquely determined by ¯ Rx(τ) except for orthogonal transformation in each eigenspace and permutation; since the eigenspaces are one-dimensional this means A is uniquely determined by equation 3 except for permutation. In addition to this separability result, A can be recovered algorithmically by simply calculating the eigenvalue decomposition of ¯ Rx(τ) (AMUSE, [3]). In practice, if the eigenvalue decomposition is problematic, a different choice of τ often resolves this problem. Nontheless, there are sources in which some components have equal autocovariances. Also, due to the fact that the autocovariance matrices are only estimated by a finite amount of samples, and due to possible colored noise, the autocovariance at τ could be badly estimated. A more general BSS algorithm called SOBI (second-order blind identification) based on time decorrelation was therefore proposed by Belouchrani et al. [4]. In addition to only diagonalizing a single autocovariance matrix, it takes a whole set of autocovariance matrices of x(t) with varying time lags τ and jointly diagonalizes the whole set. It has been shown that increasing the size of this set improves SOBI performance in noisy settings [1]. Algorithms for performing joint diagonalization of a set of symmetric commuting matrices include gradient descent on the sum of the off-diagonal terms, iterative construction of A by Givens rotation in two coordinates [7] (used in the simulations in section 4), an iterative two-step recovery of A [8] or more recently a linear least-squares algorithm for diagonalization [9], where the latter two algorithms can also search for non-orthogonal matrices A. Joint diagonalization has been used in BSS using cumulant matrices [10] or time autocovariances [4,5]. 3 Multidimensional SOBI The goal of this work is to improve SOBI performance for random processes with a higher dimensional parametrization i.e. for data sets where the random processes s and x do not depend on a single variable t, but on multiple variables (z1, . . . , zM ). A typical example is a source data set, in which each component si represents an image of size h × w. Then M = 2 and samples of s are given at z1 = 1, . . . , h, z2 = 1, . . . , w. Classically, s(z1, z2) is transformed to s(t) by fixing a mapping from the two-dimensional parameter set to the one-dimensional time parametrization of s(t), for example by concatenating columns or rows in the case of a finite number of samples. If the time structure of s(t) is not used, as in all classical ICA algorithms in which i.i.d. samples are assumed, this choice does not influence the result. However, in time-structure based algorithms such as AMUSE and SOBI results can vary greatly depending on the choice of this mapping, see figure 2. Without loss of generality we again assume centered random vectors. Then define the multidimensional covariance to be Rs(τ1, . . . , τM ) := E � s(z1 + τ1, . . . , zM + τM)s(z1, . . . , zM ) ⊤�
Chapter 6. LNCS 3195:726-733, 2004 117 4 Fabian J. Theis, Anke Meyer-Baese, and Elmar W. Lang PSfrag replacements 0.8 0.6 0.4 0.2 0 1 1d−autocov 2d−autocov 0 50 100 150 200 250 300 τ respectively |(τ1, τ2)| (rescaled to N) Fig. 1. Example of one- and two-dimensional autocovariance coefficient of the grayscale 128 × 128 Lena image after normalization to variance 1. where the expectation is taken over (z1, . . . , zM ). Rs(τ1, . . . , τM ) can be estimated given equidistant samples by replacing random variables by sample values and expectations by sums as usual. The advantage of using multidimensional autocovariances lies in the fact that now the multidimensional structure of the data set can be used more explicitly. For example, if row concatenation is used to construct s(t) from the images, horizontal lines in the image will only give trivial contributions to the autocovariance (see examples in figure 2 and section 4). Figure 1 shows the oneand two-dimensional autocovariance of the Lena image for varying τ respectively (τ1, τ2) after normalization of the image to variance 1. Clearly, the twodimensional autocovariance does not decay as quickly with increasing radius as the one-dimensional covariance. Only at multiples of the image height, the onedimensional autocovariance is significantly high i.e. captures image structure. Our contribution consists of using multidimensional autocovariances for joint diagonalization. We replace the BSS assumption of diagonal one-dimensional autocovariances by diagonal multi-dimensional autocovariances of the sources. Note that also the multidimensional covariance satisfies the equation 2. Again we assume whitened x(z1, . . . , zK). Given a autocovariance matrix ¯ Rx � τ (1) 1 , . . . , τ (1) M with n different eigenvalues, multidimensional AMUSE (mdAMUSE) detects the orthogonal unmixing mapping W by diagonalization of this matrix. In section 2, we discussed the advantages of using SOBI over AMUSE. This of course also holds in this generalized case. Hence, the multidimensional SOBI algorithm (mdSOBI ) consists of the joint diagonalization of a set of symmetrized multidimensional autocovariances � � ¯Rx τ (1) � (1) 1 , . . . , τ M , . . . , ¯ � Rx τ (K) 1 , . . . , τ (K) �� M �
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Statistical machine learning of bio
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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1.2. Uniqueness issues in independe
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1.2. Uniqueness issues in independe
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S U M M A R Y T his �le contains
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1.3. Dependent component analysis 1
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Table 1.1: BSS algorithms based on
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1.4. Sparseness 25 1.4 Sparseness O
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1.4. Sparseness 29 R 3 R 3 A BSRA R
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Part II Papers 53
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298 BIBLIOGRAPHY Barber, C., Dobkin
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300 BIBLIOGRAPHY Georgiev, P. (2001
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302 BIBLIOGRAPHY Keck, I., Theis, F
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306 BIBLIOGRAPHY Theis, F. and Inou
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