Independent Component Analysis: The Fast ICA algorithm

1. Independent Component Analysis: The Fast ICA algorithm Jonathan Kam EE 645

2. Overview • The Problem • Definition of ICA • Restrictions • Ways to solve ICA • NonGaussianity • Mutual Information • Maximum Likelihood • Fast ICA algorithm • Simulations • Conclusion

3. The Problem • Cocktail Problem • Several Sources • Several Sensors • Ex: Humans hear mixed signal, but able to unmix signals and concentrate on a sole source • Recover source signals given only mixtures • No prior knowledge of sources or mixing matrix • aka Blind Source Separation (BSS)

4. Assumptions • Source signals are statistically independent • Knowing the value of one of the components does not give any information about the others • ICs have nongaussian distributions • Initial distributions unknown • At most one Gaussian source • Recovered sources can be permutated and scaled

5. Definition of ICA • Observe N linear mixtures x1,…,xn of n independent components xj = aj1s1 + aj2s2 + … + ajnsn, for all j • aj is the column of the mixing matrix A • Assume each mixture xj and each IC sk is a random variable • Time difference between mixes dropped • Independent components are latent variables • Cannot be directly observed

6. Definition of ICA • ICA Mixture model: x=As • A is mixing matrix; s is matrix of source signals • Goal • Find some matrix W, so that s = Wx • W = inverse of A

7. Definition: Independence • Two functions independent if E{h1(y1)h2(y2)} = E{h1(y1)} E{h2(y2)} • If variables are independent, they are uncorrelated • Uncorrelated variables • Defined: E{y1y2} = E{y1} E{y2} = 0 • Uncorrelation doesn’t equal independence • Ex: (0,1),(0,-1),(1,0),(-1,0) • E{y12y22} = 0 ≠ ¼ = E{y12} E{y22} • ICA has to prove independence

8. ICA restrictions • Cannot determine variances • s and A are unknown • Scalar multipliers on s could be canceled out by a divisor on A • Multiplier could even be -1 • Cannot determine order • Order of terms can changed.

9. ICA restrictions • At most 1 Gaussian source • x1 and x2 Gaussian, uncorrelated, and unit variance • Density function is completely symmetric • Does not contain info on direction of the columns of the mixing matrix A.

10. ICA estimation • Nongaussianity estimates independent • Estimation of y = wT x • let z = AT w, so y = wTAs = zTs • y is a linear combination of si, therefore zTs is more gaussian than any of si • zTs becomes least gaussian when it is equal to one of the si • wTx = zTs equals an independent component • Maximizing nongaussianity of wTx gives us one of the independent components • Maximizing nongaussianity by measuring nongaussiantiy • Minimizing mutual information • Maximum Likelihood

11. Measuring nongaussianity • Kurtosis • Fourth order cumulant • Classical measure of nongaussianity • kurt(y) = E{y4} – 3(E{y2})2 • For gaussian y, fourth moment = 3(E{y2})2 • Kurtosis for gaussian random variables is 0 • Con – not a robust measure of nongaussianity • Sensitive to outliers

12. Measuring nongaussianity • Entropy (H): degree of information that an observation gives • A Gaussian variable has the largest entropy among all random variables of equal variance • Negentropy J • Based on the information theoretic quantity of differential entropy • Computationally difficult

13. Negentropy approximations • Classical method using higher-order moments • Validity is limited to nonrobustness of kurtosis

14. Negentropy approximations • Hyvärinen 1998b: maximum-entropy principle • G is some contrast function • v is a Gaussian variable of zero mean and unit variance • Taking G(y) = y4 makes the equation kurtosis based approximation

15. Negentropy approximations • Instead of kurtosis function, choose a contrast function G that doesn’t grow too fast Where 1≤a1≤2

16. Minimizing mutual information • Mutual information I is defined as • Measure of the dependence between random variables • I = 0 if variables are statistically independent • Equivalent to maximizing negentropy

17. Maximum Likelihood Estimation • Closely related to infomax principle • Infomax (Bell and Sejnowski, 1995) • Maximizing the output entropy of a neural network with non-linear outputs • Densities of ICs must be estimated properly • If estimation is wrong ML will give wrong results

18. Fast ICA • Preprocessing • Fast ICA algorithm • Maximize non gaussianity • Unmixing signals

19. Fast ICA: Preprocessing • Centering • Subtract its mean vector to make x a zero-mean variable • ICA algorithm does not need to estimate the mean • Estimate mean vector of s by A-1m, where m is the mean the subtracted mean

20. Fast ICA: Preprocessing • Whitening • Transform x so that its components are uncorrelated and their variances equal unity • Use eigen-value decomposition (EVD) of the covariance matrix E • D is the diagonal matrix of its eigenvalues • E is the orthogonal matrix of eigenvectors

21. Fast ICA: Preprocessing • Whitening • Transforms the mixing matrix into Ã. • Makes à orthogonal • Lessons the amount of parameters that have to be estimated from n2 to n(n-1)/2 • In large dimensions an orthogonal matrix contains approximately ½ the number of parameters

22. Fast ICA Algorithm • One-unit (component) version 1. Choose an initial weight vector w. 2. Let w+ = E{xg(wTx)} – E{g′(wTx)}w • Derivatives of contrast functions G g1(u) = tanh(a1u), g2(u) = u exp (-u2/2) 3. w = w+/||w+||. (Normalization step) 4. If not converged go back to 2 -converged if norm(wnew – wold) > ξ or norm(wold-wnew)> ξ - ξ typically around 0.0001

23. Fast ICA Algorithm • Several unit algorithm • Define B as mixing matrix and B′ as a matrix whose columns are the previously found columns of B • Add projection step before step 3 • Step 3 becomes 3. Let w(k) = w(k) - B′B′Tw(k). w = w+/||w+||

24. Simple Simulation • Separation of 2 components • Figure 1: Two independent non gaussian wav samples

25. Simple Simulation • Figure 2: Mixed signals

26. Simple Simulation • Recovered signals vs original signals Figure 3: Recovered signals Figure 4: Original signals

27. Simulation Results • IC 1 recovered in 6 steps and IC 2 recovered in 2 steps • Retested with 20000 samples • Requires approximately the same number of steps

28. Gaussian Simulation Figure 5: 2 wav samples and noise signal

29. Gaussian Simulation Figure 6: 3 mixed signals

30. Gaussian Simulation • Comparison of recovered signals vs original signals Figure 7: Recovered signals Figure 8: Original signal

31. Gaussian Simulation 2: • Tried with 2 gaussian components • Components were not estimated properly due to more than one Gaussian component Figure 10: Original signals Figure 11: Recovered signals

32. Conclusion • Fast ICA properties • No step size, unlike gradient based ICA algorithms • Finds any non-Gaussian distribution using any non linear g contrast function. • Components can be estimated one by one • Other Applications • Separation of Artifacts in image data • Find hidden factors in financial data • Reduce noise in natural images • Medical signal processing – fMRI, ECG, EEG (Mackeig)

33. References • [1] Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: Algorithms and Applications. Neural Networks Research Centre Helsinki University of Technology; Neural Networks, 13 (4-5): 411-430, 2000 • [2] Aapo Hyvärinen and Erkki Oja, A Fast Fixed-Point Algorithm for Independent Component Analysis. Helsinki University of Technology Laboratory of Computer and Information Science, Neural Computation, 9:1483:1492, 1997 • [3] Anthony J. Bell and Terrence J. Sejnowski, The ‘Independent Components’ of Natural Scenes are Edeg Filters. Howard Hughes Medical Institute Computational Neurobiology Laboratory • [4] Te-Won Lee, Mark Girolami, Terrence J. Sejnowski, Independent Component Analysis Using and Extended Infomax Algorithm for Mixed Subgaussian and Supergaussian Sources. 1997 • [5] Antti Leino, Independent Component Analysis An Overview. 2004 • [6] Erik G. Learned-Miller, John W. Fisher III, ICA Using Spacings Estimates of Entropy Journal of Machine Learning Research 4 (2003) 1271-1295.