Mathematics in Independent Component Analysis

More documents

Recommendations

Info

136 Chapter 8. Proc. NIPS 2006 5.5 5 4.5 4 3.5 3 2.5 2 (a) S2 6 5.5 5 4.5 4 3.5 3 2.5 1.5 7 8 9 10 11 12 13 2 14 3 4 5 6 7 8 1.5 9 3 3.5 4 4.5 5 5.5 6 6.5 7 5 4.5 4 3.5 3 2.5 2 1.5 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 (f) ( ˆ S1, ˆ S2) 250 200 150 100 50 (b) S3 0 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 (g) histogram of ˆ S3 5 4.5 4 3.5 3 2.5 14 13 12 11 10 9 2 (c) S4 5 4 3 2 1 0 5 4 3 2 1 0 0 1 (d) S5 8 −8 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 −7.5 −7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5 (h) S4 −3 −3.5 −4 −4.5 −5 −5.5 −6 −6.5 −7 −7.5 (i) S5 2 3 4 1 2 3 4 5 6 7 8 5 9 10 1 0 −1 −2 −3 −4 −5 0 1 2 3 4 5 6 7 8 9 10 (e) Â−1 A −1 5 −2 4 −3 3 2 −4 1 −5 0 (j) S6 Figure 3: Example application of general ISA for unknown sizes m = (1, 2, 2, 2, 3). Shown are the scatter plots i.e. densities of the source components and the mixing-separating map Â−1 A. also Cov(S) = I. Then I = Cov(X) = A Cov(S)A ⊤ = AA ⊤ so A is orthogonal. Due to the ISA assumptions, the fourth-order cross cumulants of the sources have to be trivial between different groups, and within the Gaussians. In order to find transformations of the mixtures fulfilling this property, we follow the idea of the JADE algorithmbut now in the ISA setting. We perform JBD of the (whitened) contracted quadricovariance matrices defined by Cij(X) := E � X ⊤ EijXXX ⊤� − Eij −E ⊤ ij −tr(Eij)I. Here RX := Cov(X) and Eij is a set of eigen-matrices of Cij, 1 ≤ i, j ≤ n. One simple choice is to use n 2 matrices Eij with zeros everywhere except 1 at index (i, j). More elaborate choices of eigen-matrices (with only n(n + 1)/2 or even n entries) are possible. The resulting algorithm, subspace-JADE (SJADE) not only performs NGCA by grouping Gaussians as one-dimensional components with trivial Cii’s, but also automatically finds the subspace partition m using the general JBD algorithm from section 2.3. 4 Experimental results In a first example, we consider a general ISA problem in dimension n = 10 with the unknown partition m = (1, 2, 2, 2, 3). In order to generate 2- and 3-dimensional irreducible random vectors, we decided to follow the nice visual ideas from [10] and to draw samples from a density following a known shape — in our case 2d-letters or 3d-geometrical shapes. The chosen sources densities are shown in figure 3(a-d). Another 1-dimensional source following a uniform distribution was constructed. Altogether 104 samples were used. The sources S were mixed by a mixing matrix A with coefficients uniformly randomly sampled from [−1, 1] to give mixtures X = AS. The recovered mixing matrix Â was then estimated using the above block-JADE algorithm with unknown block size; we observed that the method is quite sensitive to the choice of the threshold (here θ = 0.015). Figure 3(e) shows the composed mixing-separating system Â−1A; clearly the matrices are equal except for block permutation and scaling, which experimentally confirms theorem 1.8. The algorithm found a partition ˆm = (1, 1, 1, 2, 2, 3), so one 2d-source was misinterpreted as two 1d-sources, but by using previous knowledge combination of the correct two 1d-sources yields the original 2dsource. The resulting recovered sources ˆ S := Â−1X, figures 3(f-j), then equal the original sources except for permutation and scaling within the sources — which in the higher-dimensional cases implies transformations such as rotation of the underlying images or shapes. When applying ICA (1-ISA) to the above mixtures, we cannot expect to recover the original sources as explained in figure 1; however, some algorithms might recover the sources up to permutation. Indeed, SJADE equals JADE with additional permutation recovery because the joint block diagonalization is per-
Chapter 8. Proc. NIPS 2006 137 formed using joint diagonalization. This explains why JADE retrieves meaningful components even in this non-ICA setting as observed in [4]. In a second example, we illustrate how the algorithm deals with Gaussian sources i.e. how the subspace JADE also includes NGCA. For this we consider the case n = 5, m = (1, 1, 1, 2) and sources with two Gaussians, one uniform and a 2-dimensional irreducible component as before; 105 samples were drawn. We perform 100 Monte-Carlo simulations with random mixing matrix A, and apply SJADE with θ = 0.01. The recovered mixing matrix Â is compared with A by taking the ad-hoc measure ι(P) := �3 �2 i=1 j=1 (p2ij + p2ji ) for P := Â−1A. Indeed, we get nearly perfect recovery in 99 out of 100 runs, the median of ι(P) is very low with 0.0083. A single run diverges with ι(P) = 3.48. In order to show that the algorithm really separates the Gaussian part from the other components, we compare the recovered source kurtoses. The median kurtoses are −0.0006 ± 0.02, −0.003 ± 0.3, −1.2 ± 0.3, −1.2 ± 0.2 and −1.6 ± 0.2. The first two components have kurtoses close to zero, so they are the two Gaussians, whereas the third component has kurtosis of around −1.2, which equals the kurtosis of a uniform density. This confirms the applicability of the algorithm in the general, noisy ISA setting. 5 Conclusion Previous approaches for independent subspace analysis were restricted either to fixed group sizes or semi-parametric models. In neither case, general applicability to any kind of mixture data set was guaranteed, so blind source separation might fail. In the present contribution we introduce the concept of irreducible independent components and give an identifiability result for this general, parameter-free model together with a novel arbitrary-subspace-size algorithm based on joint block diagonalization. As in ICA, the main uniqueness theorem is an asymptotic result (but includes noisy case via NGCA). However in practice in the finite sample case, due to estimation errors the general joint block diagonality only approximately holds. Our simple solution in this contribution was to choose appropriate thresholds. But this choice is non-trivial, and adaptive methods are to be developed in future works. References [1] K. Abed-Meraim and A. Belouchrani. Algorithms for joint block diagonalization. In Proc. EUSIPCO 2004, pages 209–212, Vienna, Austria, 2004. [2] F.R. Bach and M.I. Jordan. Finding clusters in independent component analysis. In Proc. ICA 2003, pages 891–896, 2003. [3] G. Blanchard, M. Kawanabe, M. Sugiyama, V. Spokoiny, and K.-R. Müller. In search of non-gaussian components of a high-dimensional distribution. JMLR, 7:247–282, 2006. [4] J.F. Cardoso. Multidimensional independent component analysis. In Proc. of ICASSP ’98, Seattle, 1998. [5] J.F. Cardoso and A. Souloumiac. Jacobi angles for simultaneous diagonalization. SIAM J. Mat. Anal. Appl., 17(1):161–164, January 1995. [6] P. Comon. Independent component analysis - a new concept? Signal Processing, 36:287–314, 1994. [7] A. Hyvärinen and P.O. Hoyer. Emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces. Neural Computation, 12(7):1705–1720, 2000. [8] A. Hyvärinen, P.O. Hoyer, and M. Inki. Topographic independent component analysis. Neural Computation, 13(7):1525–1558, 2001. [9] J.K. Lin. Factorizing multivariate function classes. In Advances in Neural Information Processing Systems, volume 10, pages 563–569, 1998. [10] B. Poczos and A. Lörincz. Independent subspace analysis using k-nearest neighborhood distances. In Proc. ICANN 2005, volume 3696 of LNCS, pages 163–168, Warsaw, Poland, 2005. Springer. [11] F.J. Theis. Uniqueness of complex and multidimensional independent component analysis. Signal Processing, 84(5):951–956, 2004. [12] F.J. Theis and M. Kawanabe. Uniqueness of non-gaussian subspace analysis. In Proc. ICA 2006, pages 917–925, Charleston, USA, 2006. [13] R. Vollgraf and K. Obermayer. Multi-dimensional ICA to separate correlated sources. In Proc. NIPS 2001, pages 993–1000, 2001.
Page 1 and 2:
Statistical machine learning of bio
Page 3:
to Jakob
Page 6 and 7:
vi Preface
Page 8 and 9:
viii CONTENTS 3 Signal Processing 8
Page 11 and 12:
Chapter 1 Statistical machine learn
Page 13 and 14:
1.1. Introduction 5 auditory cortex
Page 15 and 16:
1.2. Uniqueness issues in independe
Page 17 and 18:
1.2. Uniqueness issues in independe
Page 19 and 20:
S U M M A R Y T his �le contains
Page 21 and 22:
1.3. Dependent component analysis 1
Page 23 and 24:
Table 1.1: BSS algorithms based on
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
1.4. Sparseness 25 1.4 Sparseness O
Page 35 and 36:
1.4. Sparseness 27 Theorem 1.4.3 (S
Page 37 and 38:
1.4. Sparseness 29 R 3 R 3 A BSRA R
Page 39 and 40:
1.4. Sparseness 31 Sparse projectio
Page 41 and 42:
1.5. Machine learning for data prep
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
1.6. Applications to biomedical dat
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
1.7. Outlook 51 noise data signal i
Page 61:
Part II Papers 53
Page 64 and 65:
56 Chapter 2. Neural Computation 16
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
82 Chapter 3. Signal Processing 84(
Page 92 and 93:
84 Chapter 3. Signal Processing 84(
Page 94 and 95: 86 Chapter 3. Signal Processing 84(
Page 96 and 97: 88 Chapter 3. Signal Processing 84(
Page 98 and 99: 90 Chapter 4. Neurocomputing 64:223
Page 112 and 113: 104 Chapter 5. IEICE TF E87-A(9):23
Page 122 and 123: 114 Chapter 6. LNCS 3195:726-733, 2
Page 132 and 133: 124 Chapter 7. Proc. ISCAS 2005, pa
Page 138 and 139: 130 Chapter 8. Proc. NIPS 2006 Towa
Page 140 and 141: 132 Chapter 8. Proc. NIPS 2006 1.3
Page 142 and 143: 134 Chapter 8. Proc. NIPS 2006 stru
Page 146 and 147: 138 Chapter 8. Proc. NIPS 2006
Page 148 and 149: 140 Chapter 9. Neurocomputing (in p
Page 164 and 165: 156 Chapter 10. IEEE TNN 16(4):992-
Page 170 and 171: 162 Chapter 11. EURASIP JASP, 2007
Page 184 and 185: 176 Chapter 12. LNCS 3195:718-725,
Page 194 and 195:
186 Chapter 13. Proc. EUSIPCO 2005
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
192 Chapter 14. Proc. ICASSP 2006 S
Page 202 and 203:
194 Chapter 14. Proc. ICASSP 2006 A
Page 204 and 205:
196 Chapter 14. Proc. ICASSP 2006
Page 206 and 207:
198 Chapter 15. Neurocomputing, 69:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
230 Chapter 16. Proc. ICA 2006, pag
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
240 Chapter 17. IEEE SPL 13(2):96-9
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
252 Chapter 19. LNCS 3195:977-984,
Page 262 and 263:
254 Chapter 19. LNCS 3195:977-984,
Page 264 and 265:
256 Chapter 19. LNCS 3195:977-984,
Page 266 and 267:
258 Chapter 19. LNCS 3195:977-984,
Page 268 and 269:
260 Chapter 19. LNCS 3195:977-984,
Page 270 and 271:
262 Chapter 20. Signal Processing 8
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
292 Chapter 21. Proc. BIOMED 2005,
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
298 BIBLIOGRAPHY Barber, C., Dobkin
Page 308 and 309:
300 BIBLIOGRAPHY Georgiev, P. (2001
Page 310 and 311:
302 BIBLIOGRAPHY Keck, I., Theis, F
Page 312 and 313:
304 BIBLIOGRAPHY Schießl, I., Sch
Page 314 and 315:
306 BIBLIOGRAPHY Theis, F. and Inou
show all

Mathematics in Independent Component Analysis

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?