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ICA and ISA Using Schweizer-Wolff Measure of Dependence

ICA and ISA Using Schweizer-Wolff Measure of Dependence. Sergey Kirshner and Barnabás Póczos Alberta Ingenuity Centre for Machine Learning, Department of Computing Science, University of Alberta, Canada. ICML 2008 Poster Session II (Paper #551). July 8, 2008.

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ICA and ISA Using Schweizer-Wolff Measure of Dependence

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  1. ICA and ISA Using Schweizer-Wolff Measure of Dependence Sergey Kirshner and Barnabás Póczos Alberta Ingenuity Centre for Machine Learning, Department of Computing Science, University of Alberta, Canada ICML 2008 Poster Session II (Paper #551) July 8, 2008

  2. Independent Component Analysis(ICA, The Cocktail Party Problem) Sources Mixing Observation Estimation 1 2 3 d y = Wx s x = As ICML 2008

  3. Find W, recover Wx Observation Sources Estimation Independent Subspace Analysis(ISA, The Woodstock Problem) ICML 2008

  4. Solving ICA • Sources can be recovered only up to scale and permutation • ICA solution: • Removing mean, E[y]=0 • Whitening, E[yyT]=I • Finding a multidimensional rotation optimizing an objective function • Sequence of 2-d Jacobi (Givens) rotations original mixed whitened rotated ICML 2008

  5. Objective Function/Contrast • Need to measure independence (ICA) • Mutual information (MI) estimation • Kernel-ICA [Bach & Jordan, 2002] • Entropy estimation • RADICAL [Learned-Miller & Fisher, 2003] • FastICA [Hyvarinen, 1999] • ML estimation • KDICA [Chen, 2006] • Moment-based methods • JADE [Cardoso, 1993] • Correlation-based methods • [Jutten and Herault, 1991] • Need to measure dependence (ISA) • Grouping signals recovered by ICA • Cardoso’s conjecture [Cardoso, 1998] ICML 2008

  6. Contribution: New ICA Contrast • Using a measure of dependence based on copulas, joint distributions over ranks • Properties • Does not require density estimation • Non-parametric • Advantages • Very robust to outliers • Frequently performs as well as state of the art algorithms (and sometimes better) • Contrast can be used with ISA • Can be used to estimate dependence • Easy to implement (code publicly available) • Disadvantages • Somewhat slow (although not prohibitively so) • Needs more samples to demix near-Gaussian sources ICML 2008

  7. Ranks ICML 2008

  8. Ranks • Invariant under monotonic transformations • Implied non-linearity • Not very sensitive to outliers ICML 2008

  9. Probability Integral Transform ICML 2008

  10. Distribution Over Ranks ICML 2008

  11. Copula Bivariate copula C is a multivariate distribution (cdf) defined on a unit square with uniform univariate marginals: ICML 2008

  12. Sklar’s Theorem [Sklar, 59] = + ICML 2008

  13. Useful Properties of Copulas • Preserve rank correlations between the variables • Preserve mutual information • Can be viewed as a canonical form of a multivariate distribution for the purpose of the estimation of multivariate dependence ICML 2008

  14. Independence Copula ICML 2008

  15. Schweizer-Wolff Measures of Dependence • L-norm between the copula for the distribution and the independence copula • Range is [0,1] • 0 if and only if variables are independent • Examples • s (L1-norm) • k (L∞-norm) [Schweizer and Wolff, 81] ICML 2008

  16. Empirical Copula [Deheuvels,79] ICML 2008

  17. Using Empirical Copula • Useful for computing sample versions of functions on copulas ICML 2008

  18. Schweizer-Wolff ICA (SWICA) • Inputs:X, a 2 x N matrix of signals, K,number of evaluation angles • For q=0,p/(2K),…,(K-1)p/(2K) • Compute rotation matrix • Compute rotated signals Y(q)=W(q)X. • Compute s(Y(q)), a sample estimate of s • Sort • Compute s • Find best angle qm=argminqs(Y(q)) • Output: Rotation matrix W=W(qm), demixed signal Y=Y(qm), and estimated dependence measure s=s(Y(qm)) O(1) O(N) O(NlogN) O(NlogB) • Sort into B bins O(N+B2) O(N2) • Compute B-bin approximation to s O(K) O(KNlogB) O(KN2) ICML 2008

  19. demixing mixing Amari Error • Measures how close a square matrix is to a permutation matrix B = WA ICML 2008

  20. Synthetic Marginal Distributions [Bach and Jordan 02] ICML 2008

  21. ICA Comparison (d=2, N=1000, 1000 repetitions) ICML 2008

  22. Outliers(d=2, N=1000, 10000 repetitions) ICML 2008

  23. Outliers: Multidimensional Comparison d=8, N=5000, 100 repetitions d=4, N=2000, 1000 repetitions Amari error Number of post-whitening outliers ICML 2008

  24. Unmixing Image Sources with Outliers (3% outliers) Original SWICA FastICA Mixed [http://www.cis.hut.fi/projects/ica/data/images/] ICML 2008

  25. Independent Subspace Analysis ICML 2008

  26. Summary • New contrast based on a measure of dependence for distribution over ranks • Robust to outliers • Comparable performance to state of the art algorithms • Can handle a moderate number of sources (d=20) • Can be used to solve ISA • Future work • Further acceleration of SWICA • What types of sources does SWICA do well on and why? • Non-parametric estimation of mutual information ICML 2008

  27. Software http://www.cs.ualberta.ca/~sergey/SWICA ICML 2008

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