Independent Component Analysis

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Independent Component Analysis

  1. 1. . . Independent Component Analysis for Blind Source Separation . .. . . Tatsuya Yokota Tokyo Institute of Technology Jan. 31, 2012Jan. 31, 2012 1/28
  2. 2. Outline . . . Blind Source Separation 1 . . . Independent Component Analysis 2 . . . Experiments 3 . . . Summary 4Jan. 31, 2012 2/28
  3. 3. What’s a Blind Source Separation Blind Source Separation is a method to estimate original signals from observed signals which consist of mixed original signals and noise.Jan. 31, 2012 3/28
  4. 4. Example of BSS BSS is often used for Speech analysis and Image analysis.Jan. 31, 2012 4/28
  5. 5. Example of BSS (cont’d) BSS is also very important for brain signal analysis.Jan. 31, 2012 5/28
  6. 6. Model Formalization The problem of BSS is formalized as follow: The matrix X ∈ Rm×d (1) denotes original signals, where m is number of original signals, and d is dimension of one signal. We consider that the observed signals Y ∈ Rn×d are given by linear mixing system as Y = AX + E, (2) where A ∈ Rn×m is the unknown mixing matrix and E ∈ Rn×d denotes a noise. Basically, n ≥ m. ˆ ˆ ˆ The goal of BSS is to estimate A and X so that X provides unknown original signal as possible.Jan. 31, 2012 6/28
  7. 7. Kinds of BSS Methods Actually, degree of freedom of BSS model is very high to estimate A and X. Because there are a huge number of combinations (A, X) which satisfy Y = AX + E. Therefore, we need some constraint to solve the BSS problem such as: PCA : orthogonal constraint SCA : sparsity constraint NMF : non-negativity constraint ICA : in-dependency constraint In this way, there are many methods to solve the BSS problem depending on the constraints. What we use is depend on subject matter. The Non-negative Matrix Factorization(NMF) was introduced in my previous seminar. We can get its solution by the alternating least squares algorithm. Today, I will introduce another method the Independent Component Analysis.Jan. 31, 2012 7/28
  8. 8. Independent Component Analysis . The Cocktail Party Problem . .. x1 (t) = a11 s1 (t) + a12 s2 (t) + a13 s3 (t) (3) x2 (t) = a21 s1 (t) + a22 s2 (t) + a23 s3 (t) (4) . x3 (t) = a31 s1 (t) + a32 s2 (t) + a33 s3 (t) (5) .. . . x is an observed signal, and s is an original signal. We assume that {s1 , s2 , s3 } are statistically independent of each other. . The model of ICA . .. Independent Component Analysis (ICA) is to estimate the independent components s(t) from x(t). . x(t) = As(t) (6) .. . .Jan. 31, 2012 8/28
  9. 9. Approach . Hypothesis of ICA . .. ... {si } are statistically independent of each other, 1 p(s1 , s2 , . . . , sn ) = p(s1 )p(s2 ) · · · p(sn ). (7) ... 2 {si } follow the Non-Gaussian distribution. If {si } follows the Gaussian distribution, then ICA is impossible. ... A is a regular matrix. 3 Therefore, we can rewrite the model as s(t) = Bx(t), (8) where B = A−1 . It is only necessary to estimate B so that {si } are . independent. .. . .Jan. 31, 2012 9/28
  10. 10. Whitening and ICA . Definition of White signal . .. White signals are defined as any z which satisfies conditions of . E[z] = 0, E[zz T ] = I. (9) .. . . First, we show an example of original independent signals and observed signal as follow: (a) source (s1 , s2 ) (b) observed (x1 , x2 ) Observed signals x(t) are given by x(t) = As(t). ICA give us the original signals s(t) by s(t) = Bx(t).Jan. 31, 2012 10/28
  11. 11. Whitening and ICA (cont’d) Whitening is useful for preprocessing of ICA. First, we apply the whitening to observed signals x(t). (c) observed (x1 , x2 ) (d) whitening (z1 , z2 ) The whitening signals are denoted as (z1 , z2 ), and they are given by z(t) = V x(t), (10) where V is a whitening matrix for x. Model becomes s(t) = U z(t) = U V x(t) = Bx(t), (11) and U is an orthogonal transform matrix. We can say that the whitening simplifies the ICA problem. So it is only necessary to estimate U .Jan. 31, 2012 11/28
  12. 12. Non-Gaussianity and ICA Non-Gaussianity is a measure of in-dependency. According to the central limit theorem, the Gaussianity of x(t) must be larger than s(t). Now, we put bT as mixing vector, si (t) = bT x(t). We want to maximize the i ˆ i Non-Gaussianity of (bT x(t)). Then such b is a part of solution B. i For example, there are following two vector b and b. We can say that b is better than b .Jan. 31, 2012 12/28
  13. 13. Maximization of Kurtosis Kurtosis is a measures of Non-Gaussianity. Kurtosis is defined by kurt(y) = E[y 4 ] − 3(E[y 2 ])2 . (12) We assume that y is white (i.e. E[y] = 0, E[y 2 ] = 1 ), then kurt(y) = E[y 4 ] − 3. (13) We can solve the ICA problem by ˆ b = max |kurt(bT x(t))|. (14) bJan. 31, 2012 13/28
  14. 14. Fast ICA algorithm based on Kurtosis We consider z is a white signal given from x. And we consider to maximize the absolute value of kurtosis as maximize |kurt(wT z)|, s.t. wT w = 1. (15) Differential of |kurt(wT z)| is given by ∂|kurt(wT z)| ∂ = E{(wT z)4 } − 3E{(wT z)2 }2 (16) ∂w ∂w ∂ = E{(wT z)4 } − 3{||w||2 }2 (because E(zz T ) = I) (17) ∂w = 4sign[kurt(wT z)] E{z(wT z)3 } − 3w||w||2 (18)Jan. 31, 2012 14/28
  15. 15. Fast ICA algorithm based on Kurtosis (cont’d) According to the gradient method, we can obtain following algorithm: . Gradient algorithm based on Kurtosis . .. w ← w + ∆w, (19) w w← , (20) ||w|| . ∆w ∝ sign[kurt(wT z)] E{z(wT z)3 } − 3w . (21) .. . . We can see that above algorithm converge when w ∝ ∆w. And w and −w are equivalent solution, so we can obtain another algorithm: . Fast ICA algorithm based on Kurtosis . .. w ← E{z(wT z)3 } − 3w, (22) w w← . (23) . ||w|| .. . . It is well known as a fast convergence algorithm for ICA !!Jan. 31, 2012 15/28
  16. 16. Example 3 4 2 2 1 0 0 -1 -2 -2 -4 -3-3 -2 -1 0 1 2 3 -4 -2 0 2 4 (a) subgaussian (b) supergaussian Figure: Example of ICAJan. 31, 2012 16/28
  17. 17. Issue of Kurtosis Kurtosis has a fatal issue that it is very weak with the outliers. Because Kurtosis is a fourth order function. Following figure depicts the result of kurtosis based ICA with outlier. The rates of outliers is only 2 %. 4 3 2 1 0 -1 -2 -3 -4-4 -3 -2 -1 0 1 2 3 4 Figure: With outliers (20 : 1000)Jan. 31, 2012 17/28
  18. 18. Neg-entropy based ICA Kurtosis is very weak with outliers. Hence, the Neg-entropy is often used for ICA. In strictly, the approximation of neg-entropy is often used, because it is robust for outliers. Neg-entropy is defined by J(y) = H(yGauss ) − H(y), (24) where H(y) = − py (η) log py (η)dη, (25) and yGauss is a Gaussian distribution of µ = E(y) and σ = E((y − µ)2 ). If y follows Gaussian distribution, then J(y) = 0.Jan. 31, 2012 18/28
  19. 19. Fast ICA algorithm based on Neg-entropy The approximation procedure of neg-entropy is complex, then it is omitted here. We just introduce the fast ICA algorithm based on neg-entropy: . Fast ICA algorithm based on Neg-entropy . .. w ← E[zg(wT z)] − E[g (wT z)]w (26) w w← (27) . ||w|| .. . . where we can select functions g and g from . .. g1 (y) = tanh(a1 y) and g1 (y) = a1 (1 − tanh2 (a1 y)), 1 . .. g2 (y) = y exp(−y 2 /2) and g (y) = (1 − y 2 ) exp(−y 2 /2), 2 2 ... 3 g3 (y) = y 3 and g3 (y) = 3y 2 . 1 ≤ a1 ≤ 2. Please note that (g3 , g3 ) is equivalent to Kurtosis based ICA.Jan. 31, 2012 19/28
  20. 20. Examples We can see that neg-entropy based ICA is robust for outliers. 4 4 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -4-4 -3 -2 -1 0 1 2 3 4 -4-4 -3 -2 -1 0 1 2 3 4 (a) Kurtosis based (b) Neg-entropy based (using g1 ) Figure: With outliers (20 : 1000)Jan. 31, 2012 20/28
  21. 21. Experiments: Real Image 1 (a) ob 1 (b) ob 2 (a) newyork (a) estimated signal 1 Figure: Observed Signals (b) shanghai (b) estimated signal 2 Figure: Original Signals Figure: Estimated SignalsJan. 31, 2012 21/28
  22. 22. Experiments: Real Image 2 (a) ob 1 (b) ob 2 (a) buta (a) estimated signal 1 Figure: Observed Signals (b) kobe (b) estimated signal 2 Figure: Original Signals Figure: Estimated SignalsJan. 31, 2012 22/28
  23. 23. Experiments: Real Image 2 (using filtering) (a) ob 1 (b) ob 2 (a) buta (a) estimated signal 1 Figure: Observed Signals (b) kobe (b) estimated signal 2 Figure: Original Signals Figure: Estimated SignalsJan. 31, 2012 23/28
  24. 24. Experiments: Real Image 3 (using filtering) (a) nyc (b) sha (c) rock (d) pig (a) estimated signal 1 (b) estimated signal 2 (e) obs1 (f) obs2 (c) estimated signal 3 (d) estimated signal 4 (g) obs3 (h) obs4 Figure: Estimated Signals Figure: Ori. & Obs.Jan. 31, 2012 24/28
  25. 25. Approaches of ICA In this research area, many method for ICA are studied and proposed as follow: . .. Criteria of ICA [Hyv¨rinen et al., 2001] 1 a Non-Gaussianity based ICA* Kurtosis based ICA* Neg-entropy based ICA* MLE based ICA Mutual information based ICA Non-linear ICA Tensor ICA ... 2 Solving Algorithm for ICA gradient method* fast fixed-point algorithm* [Hyv¨rinen and Oja, 1997] a (‘*’ were introduced today.)Jan. 31, 2012 25/28
  26. 26. Summary I introduced about BSS problem and basic ICA techniques (Kurtosis, Neg-entropy). Kurtosis is weak with outliers. Neg-entropy is proposed as a robust measure of Non-Gaussianity. I conducted experiments of ICA using Image data. In some case, worse results are obtained. But I solved this issue by using differential filter. This technique is proposed in [Hyv¨rinen, 1998]. a We knew that the differential filter is very effective for ICA.Jan. 31, 2012 26/28
  27. 27. Bibliography I [Hyv¨rinen, 1998] Hyv¨rinen, A. (1998). a a Independent component analysis for time-dependent stochastic processes. [Hyv¨rinen et al., 2001] Hyv¨rinen, A., Karhunen, J., and Oja, E. (2001). a a Independent Component Analysis. Wiley. [Hyv¨rinen and Oja, 1997] Hyv¨rinen, A. and Oja, E. (1997). a a A fast fixed-point algorithm for independent component analysis. Neural Computation, 9:1483–1492.Jan. 31, 2012 27/28
  28. 28. Thank you for listeningJan. 31, 2012 28/28

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