Mathematics in Independent Component Analysis

More documents

Recommendations

Info

130 Chapter 8. Proc. NIPS 2006 Towards a general independent subspace analysis Fabian J. Theis Max Planck Institute for Dynamics and Self-Organisation & Bernstein Center for Computational Neuroscience Bunsenstr. 10, 37073 Göttingen, Germany fabian@theis.name Abstract The increasingly popular independent component analysis (ICA) may only be applied to data following the generative ICA model in order to guarantee algorithmindependent and theoretically valid results. Subspace ICA models generalize the assumption of component independence to independence between groups of components. They are attractive candidates for dimensionality reduction methods, however are currently limited by the assumption of equal group sizes or less general semi-parametric models. By introducing the concept of irreducible independent subspaces or components, we present a generalization to a parameter-free mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by including previous results on non-Gaussian component analysis. After introducing this general model, we discuss joint block diagonalization with unknown block sizes, on which we base a simple extension of JADE to algorithmically perform the subspace analysis. Simulations confirm the feasibility of the algorithm. 1 Independent subspace analysis A random vector Y is called an independent component of the random vector X, if there exists an invertible matrix A and a decomposition X = A(Y,Z) such that Y and Z are stochastically independent. The goal of a general independent subspace analysis (ISA) or multidimensional independent component analysis is the decomposition of an arbitrary random vector X into independent components. If X is to be decomposed into one-dimensional components, this coincides with ordinary independent component analysis (ICA). Similarly, if the independent components are required to be of the same dimension k, then this is denoted by multidimensional ICA of fixed group size k or simply k-ISA. So 1-ISA is equivalent to ICA. 1.1 Why extend ICA? An important structural aspect in the search for decompositions is the knowledge of the number of solutions i.e. the indeterminacies of the problem. Without it, the result of any ICA or ISA algorithm cannot be compared with other solutions, so for instance blind source separation (BSS) would be impossible. Clearly, given an ISA solution, invertible transforms in each component (scaling matrices L) as well as permutations of components of the same dimension (permutation matrices P) give again an ISA of X. And indeed, in the special case of ICA, scaling and permutation are already all indeterminacies given that at most one Gaussian is contained in X [6]. This is one of the key theoretical results in ICA, allowing the usage of ICA for solving BSS problems and hence stimulating many applications. It has been shown that also for k-ISA, scalings and permutations as above are the only indeterminacies [11], given some additional rather weak restrictions to the model. However, a serious drawback of k-ISA (and hence of ICA) lies in the fact that the requirement fixed group-size k does not allow us to apply this analysis to an arbitrary random vector. Indeed,
Chapter 8. Proc. NIPS 2006 131 crosstalking error 4 3 2 1 0 FastICA JADE Extended Infomax Figure 1: Applying ICA to a random vector X = AS that does not fulfill the ICA model; here S is chosen to consist of a two-dimensional and a one-dimensional irreducible component. Shown are the statistics over 100 runs of the Amari error of the random original and the reconstructed mixing matrix using the three ICA-algorithms FastICA, JADE and Extended Infomax. Clearly, the original mixing matrix could not be reconstructed in any of the experiments. However, interestingly, the latter two algorithms do indeed find an ISA up to permutation, which will be explained in section 3. theoretically speaking, it may only be applied to random vectors following the k-ISA blind source separation model, which means that they have to be mixtures of a random vector that consists of independent groups of size k. If this is the case, uniqueness up to permutation and scaling holds as noted above; however if k-ISA is applied to any random vector, a decomposition into groups that are only ‘as independent as possible’ cannot be unique and depends on the contrast and the algorithm. In the literature, ICA is often applied to find representations fulfilling the independence condition as well as possible, however care has to be taken; the strong uniqueness result is not valid any more, and the results may depend on the algorithm as illustrated in figure 1. This work aims at finding an ISA model that allows applicability to any random vector. After reviewing previous approaches, we will provide such a model together with a corresponding uniqueness result and a preliminary algorithm. 1.2 Previous approaches to ISA for dependent component analysis Generalizations of the ICA model that are to include dependencies of multiple one-dimensional components have been studied for quite some time. ISA in the terminology of multidimensional ICA has first been introduced by Cardoso [4] using geometrical motivations. His model as well as the related but independently proposed factorization of multivariate function classes [9] is quite general, however no identifiability results were presented, and applicability to an arbitrary random vector was unclear; later, in the special case of equal group sizes (k-ISA) uniqueness results have been extended from the ICA theory [11]. Algorithmic enhancements in this setting have been recently studied by [10]. Moreover, if the observation contain additional structures such as spatial or temporal structures, these may be used for the multidimensional separation [13]. Hyvärinen and Hoyer presented a special case of k-ISA by combining it with invariant feature subspace analysis [7]. They model the dependence within a k-tuple explicitly and are therefore able to propose more efficient algorithms without having to resort to the problematic multidimensional density estimation. A related relaxation of the ICA assumption is given by topographic ICA [8], where dependencies between all components are assumed and modelled along a topographic structure (e.g. a 2-dimensional grid). Bach and Jordan [2] formulate ISA as a component clustering problem, which necessitates a model for inter-cluster independence and intra-cluster dependence. For the latter, they propose to use a tree-structure as employed by their tree dependepent component analysis. Together with inter-cluster independence, this implies a search for a transformation of the mixtures into a forest i.e. a set of disjoint trees. However, the above models are all semi-parametric and hence not fully blind. In the following, no additional structures are necessary for the separation.
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: 80 Chapter 2. Neural Computation 16
Page 90 and 91: 82 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 140 and 141: 132 Chapter 8. Proc. NIPS 2006 1.3
Page 142 and 143: 134 Chapter 8. Proc. NIPS 2006 stru
Page 144 and 145: 136 Chapter 8. Proc. NIPS 2006 5.5
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 186 and 187: 178 Chapter 12. LNCS 3195:718-725,
Page 188 and 189:
180 Chapter 12. LNCS 3195:718-725,
Page 190 and 191:
182 Chapter 12. LNCS 3195:718-725,
Page 192 and 193:
184 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

Create successful ePaper yourself

Delete template?

Save as template?