Mathematics in Independent Component Analysis

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72 Chapter 2. Neural Computation 16:1827-1850, 2004 A New Concept for Separability Problems in BSS 1843 hence, E = (σ 2 ν) −4 � m � � � ϕ(x − x (k) )ϕ(x − x (l) )(x (k) i i
Chapter 2. Neural Computation 16:1827-1850, 2004 73 1844 F. Theis Note that E represents a new approximate measure of independence. Therefore, the linear BSS algorithm can now be readily generalized to nonlinear situations by finding an appropriate parameterization of the possibly nonlinear separating model. The proposed algorithm basically performs a global diagonalization of the logarithmic Hessian after prewhitening. Interestingly, this is similar to traditional BSS algorithms based on joint diagonalization such as JADE (Cardoso & Souloumiac, 1993) using cumulant matrices or AMUSE (Tong, Liu, Soan, & Huang, 1991) and SOBI (Belouchrani, Meraim, Cardoso, & Moulines, 1997) employing time decorrelation. Instead of using a global energy function as proposed above, we could therefore also jointly diagonalize a given set of Hessians (respectively, separator matrices, as above; see also Yeredor, 2000). Another relation to previously proposed ICA algorithms lies in the kernel approximation technique. Gaussian or generalized gaussian kernels have already been used in the field of independent component analysis to model the source densities (Lee & Lewicki, 2000; Habl, Bauer, Putonet, Rodriguez-Alvarez, & Lang, 2001), thus giving an estimate of the score function used in Bell-Sejnowski-type semiparametric algorithms (Bell, & Sejnowski, 1995) or enabling direct separation using a maximum likelihood parameter estimation. Our algorithm also uses density approximation, but employs this for the mixture density, which can be problematic in higher dimensions. A different approach not involving density approximation is a direct sample-based Hessian estimation similar to Lin (1998). 5 Separability of Postnonlinear BSS In this section, we show how to use the idea of Hessian diagonalization in order to give separability proofs in nonlinear situations, more precisely, in the setting of postnonlinear BSS. After stating the postnonlinear BSS model and the general (to the knowledge of the author, not yet proven) separability theorem, we will prove postnonlinear separability in the case of random vectors with distributions that are somewhere locally constant and nonzero (e.g., uniform distributions). A possible proof of postnonlinear separability has been suggested by Taleb and Jutten (1999); however, the proof applies only to densities with at least one zero and furthermore contains an error rendering the proof applicable only to restricted situations. Definition 3. A function f : R n → R n is called diagonal if each component fi(x) of f(x) depends on only the variable xi. In this case, we often omit the other variables and write f(x1,...,xn) = ( f1(x1),..., fn(xn)); sof ≡ f1 ×···× fn where × denotes the Cartesian product.
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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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1.3. Dependent component analysis 1
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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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Mathematics in Independent Component Analysis

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