Mathematics in Independent Component Analysis

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232 Chapter 16. Proc. ICA 2006, pages 917-925 1.2 Uniqueness theorem Definition 1. X=AS with A∈Gl(d), S=(SN, SG) and SN∈ L2(Ω,R n ) is called an n-decomposition of X if SN and SG are stochastically independent and SG is Gaussian. In this case, X is said to be n-decomposable. Hence an n-decomposition of X corresponds to the NGSA problem. If as before A=(AN, AG), then the n-dimensional subvectorspace im(AN)⊂R d is called the non- Gaussian subspace, and im(AG) the Gaussian subspace of the decomposition; here im(A) denotes the image of the linear map A. Definition 2. X is denoted to be minimally n-decomposable if X is not (n−1)-decomposable. Then dime(X) := n is called the essential dimension of X. For example, the essential dimension dime(X) is zero if and only if X is Gaussian, whereas the essential dimension of a d-dimensional mutually independent Laplacian is d. The following theorem is the main theoretical contribution of this work. It essentially connects uniqueness of the dimension reduction model with minimality, and gives a simple characterization for it. Theorem 1 (Uniqueness of NGSA). Let n
Chapter 16. Proc. ICA 2006, pages 917-925 233 1.3 Proof of theorem 1 First note that the theorem holds trivially for n=0, because in this case X is Gaussian. So in the following let 0
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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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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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298 BIBLIOGRAPHY Barber, C., Dobkin
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300 BIBLIOGRAPHY Georgiev, P. (2001
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306 BIBLIOGRAPHY Theis, F. and Inou
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Mathematics in Independent Component Analysis

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