Mathematics in Independent Component Analysis

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236 Chapter 16. Proc. ICA 2006, pages 917-925 where the fourth-order cumulant tensor is defined as cum(Xi, X j, Xk, Xl) := E(XiX jXkXl)− E(XiX j)E(XkXl)−E(XiXk)E(X jXl)−E(XiXl)E(X jXK). The deviation (ii) from Gaussianity of XG can simply be measured by kurtosis, which in the case of white X means δG(X) := 1 (n−d) d� j=n+1 � � �E(X 4 i )−3� � �. Altogether, we can therefore define a total model deviation as the weighted sum of the above indices; the weight in the following was chosen experimentally to approximately yield even contributions of the two measures: δ(X)=10n(d−n)δI(X)+δG(X) For numerical tests, we generate two different non-Gaussian source data sets, see figure 1(d) and also [4], figure 1. The first source set (I) is an n-dimensional dependent sub-Gaussian random vector given by an isotropic uniform density within the unit disc, and source set (II) a 2-dimensional dependent super- and sub-Gaussian, given by p(s1, s2)∝exp(−|s1|)1[c(s1),c(s1)+1], where c(s1)=0 if|s1|≤ln 2 and c(s1)=−1 otherwise. Normalization was chosen to guarantee Cov(S N)=Iin advance. In order to test for model violations, we have to find two representations X=AS and X=A ′ S ′ of the same mixtures. After multiplication by A −1 we may as before assume that a single representation X=AS is given with X and S both fulfilling the dimension reduction model (1), and we have to show that ANG = AGN = 0 if the decomposition is minimal (corollary 1). The latter can be tested numerically by using the so-called normalized crosserror E(A) := 1 2n(d−n) � �ANG� 2 F +�AGN� 2 F where�.�F is some matrix norm, in our case the Frobenius norm. In order to reduce the d 2 -dimensional search space, after whitening we may assume that A∈O(d), so only d(d− 1)/2 dimensions have to be searched. O(d) can be uniformly sampled for example by choosing B with Gaussian i.i.d. coefficients and orthogonalizing A := (BB ⊤ ) −1/2 B. We perform 10 4 Monte-Carlo runs with random A∈O(d). Sources have been generated with T= 10 4 samples, n-dimensional non-Gaussian part (a) and (b) from above, and (d−n)-dimensional i.i.d. Gaussians. We measure model deviationδ(AS) and compare it with the deviation E(A) from block-diagonality. The results for varying parameters are given in figure 1(a-c). In all three cases we observe that the smaller the model deviation, the smaller also the crosserror. This gives an asymptotic confirmation of corollary 1, indicating that by random sampling no nonuniqueness realizations have been found. 3 Conclusion By minimality of the decomposition (1), we gave a necessary condition for the uniqueness of non-Gaussian subspace analysis. Together with the assumption of existing covariance, this was already sufficient to guarantee model uniqueness. Our result allows NGSA algorithms to find the unknown, unique signal space within a noisy highdimensional data set [6]. � ,
Chapter 16. Proc. ICA 2006, pages 917-925 237 model deviationδ(AS) model deviationδ(AS) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 2.5 2 1.5 1 0.5 crosserror E(A) (a) n=2, d= 5, source (I) 0 0 0.1 0.2 0.3 0.4 0.5 model deviationδ(AS) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 crosserror E(A) (b) n=3, d=5, source (I) crosserror E(A) (c) n=2, d=4, source (II) (d) Laplacian & uniform source Fig. 1. (a-c): total model deviationδ(AS) of the transformed sources versus crosserror E(A) of the mixing matrix for 10 4 Monte-Carlo runs. The circle◦indicates the actual source model deviation (non-zero due to finite sample sizes). (d): 2-dimensional dependent sub-Gaussian source (II). References 1. Friedman, J., Tukey, J.: A projection pursuit algorithm for exploratory data analysis. IEEE Trans. on Computers 23 (1975) 881–890 2. Hyvärinen, A., Karhunen, J., Oja, E.: Independent component analysis. John Wiley & Sons (2001) 3. Blanchard, G., Kawanabe, M., Sugiyama, M., Spokoiny, V., Müller, K.R.: In search of nongaussian components of a high-dimensional distribution. JMLR (2005) In revision. The preprint is available at http://www.cs.titech.ac.jp/ tr/reports/2005/TR05-0003.pdf. 4. Kawanabe, M.: Linear dimension reduction based on the fourth-order cumulant tensor. In: Proc. ICANN 2005. Volume 3697 of LNCS., Warsaw, Poland, Springer (2005) 151–156 5. Theis, F.: Uniqueness of complex and multidimensional independent component analysis. Signal Processing 84 (2004) 951–956 6. Kawanabe, M., Theis, F.: Extracting non-gaussian subspaces by characteristic functions. In: submitted to ICA 2006. (2006) 7. Comon, P.: Independent component analysis - a new concept? Signal Processing 36 (1994) 287–314 8. Theis, F.: A new concept for separability problems in blind source separation. Neural Computation 16 (2004) 1827–1850 9. Theis, F.: Multidimensional independent component analysis using characteristic functions. In: Proc. EUSIPCO 2005, Antalya, Turkey (2005)
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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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