Matrix information geometry

Cover......Page 1
Matrix Information \rGeometry......Page 3
Welcome to ‘‘Matrix Information Geometry’’......Page 6
3 Organization......Page 7
4 Sponsors......Page 8
Contents......Page 10
17 Particle Filtering on Riemannian Manifolds. Application to Covariance Matrices Tracking......Page 12
1.1.1 State-of-the-Art......Page 15
1.1.2 Aim of the Study and Chapter Organisation......Page 16
1.2.1 Partial Ordering vs. Total Ordering......Page 17
1.2.2 Lexicographic Total Orderings Based  on Tensor Invariants......Page 19
1.2.3 Lexicographic Total Orderings Based  on Prior Sets (B , F)......Page 22
1.3 Partial Spectral Ordering for PDS Matrices  and Inverse Eigenvalue Problem......Page 24
1.3.1 Spectral Sup/Inf on Geometric Mean Basis......Page 25
1.3.2 Spectral Sup/Inf on Optimized Basis......Page 26
1.4.1 Counter-Harmonic Mean......Page 28
1.4.2 Counter-Harmonic Mean for PDS Matrices......Page 29
1.5 Application to Nonlinear Filtering of Matrix-Valued  Images......Page 35
1.6 Conclusions and Perspectives......Page 39
References......Page 40
2.2 The Binary Geometric Mean......Page 43
2.3 The Differential Geometry Connection......Page 47
2.4 The ALM Mean......Page 48
2.5 The Riemannian Mean......Page 50
2.6 Reaching the Riemannian Barycentre......Page 52
2.7 Summary......Page 57
References......Page 58
3.1 Introduction......Page 60
3.2 Contraction Properties of the Kalman Filter......Page 62
3.2.1 Natural Metric of P+(n)......Page 63
3.2.2 Contraction Property......Page 64
3.3 Low-Rank Kalman Filtering......Page 67
3.5 Contraction Properties of the Low-Rank Kalman Filter......Page 69
3.5.1 Local Contraction of Subspace Flows......Page 70
3.5.2 Local Contraction of Low-Rank Riccati Flow......Page 71
3.5.4 A Counter-Example for Skew-Symmetric A......Page 72
3.6 Conclusion......Page 73
References......Page 74
4.1 Introduction......Page 76
4.2 KV Complex and Statistical Structures......Page 78
4.2.1 H2(A) and the Rigidity Problem......Page 81
4.2.2 Rigidity of Locally Flat Structures and Local Convexity......Page 83
4.3 The Completeness Problem......Page 84
4.4 Twisted Cohomology. Transverse Statistical Structures......Page 88
4.5 Duality. Comparison criteria......Page 93
4.5.1 Duality in Riemannian Manifolds......Page 94
4.5.2 Maurer-Cartan Polynomial and the Comparison Problem......Page 95
4.5.3 Comparison Criteria for Statistical Models......Page 97
References......Page 98
5.1 Introduction......Page 100
5.2 Determinant......Page 104
5.3 Permanent......Page 107
5.4 Tensor Power......Page 110
5.5 Antisymmetric Tensor Power......Page 111
5.6 Symmetric Tensor Power......Page 112
5.7 Coefficients of Characteristic Polynomial......Page 113
References......Page 116
6.1 Introduction and Literature Review......Page 117
6.1.2 Log-Euclidean Mean......Page 118
6.2 Divergences from Jensen Convexity Gaps......Page 119
6.3 Concave--Convex Minimization for Jensen-Based Matrix Means......Page 122
6.4 Matrix α-Log-Det Divergence......Page 123
6.5.1 Synthetic Dataset......Page 124
6.5.2 Shape Clustering......Page 125
References......Page 127
7.1 Introduction......Page 129
7.2.1 Basics of Lie Groups......Page 132
7.2.2 Lie Group Exponential and Logarithm......Page 134
7.2.3 Affine Connection Spaces......Page 136
7.2.4 Canonical Cartan Connection and Group Geodesics......Page 138
7.3 Bi-Invariant Means in Lie Groups......Page 141
7.3.1 Bi-Invariant Metrics on Lie Groups......Page 142
7.3.2 There Is No Bi-Invariant Metric for Rigid Transformations......Page 144
7.3.3 A Barycentric Definition of the Mean?......Page 146
7.3.4 A Fixed Point Iteration to Compute the Karcher Mean......Page 147
7.3.5 Bi-Invariant Means with Exponential Barycenters......Page 149
7.3.6 Existence of the Bi-Invariant Mean......Page 151
7.3.7 The Barycentric Fixed Point Iteration Is Converging......Page 152
7.4.1 Scalings and Translations in n-D......Page 154
7.4.2 The Heisenberg Group......Page 157
7.4.3 Scaled Upper Unitriangular Matrix Group......Page 159
7.4.4 General Rigid-Body Transformations......Page 160
7.4.5 2D Rigid Transformations......Page 165
7.4.6 General Linear Transformations......Page 166
7.5 Perspectives......Page 168
References......Page 170
8.1 Introduction......Page 173
8.2 Our Motivation: Radar Target Detection......Page 177
8.3 Riemannian Median and its Estimation......Page 178
8.4.1 Consistency of Fréchet Medians in Metric Spaces......Page 181
8.4.2 Robustness of Fréchet Medians in Riemannian Manifolds......Page 182
8.4.3 Uniqueness of Fréchet Sample Medians in Compact Riemannian Manifolds......Page 184
8.5 Stochastic and Deterministic Algorithms for Computing Means of Probability Measures......Page 185
8.5.1 Stochastic Algorithms for Computing p-Means......Page 186
8.5.3 Computing p-Means by Gradient Descent......Page 189
8.6 Riemannian Geometry of Toeplitz Covariance Matrices and Applications......Page 191
8.6.1 Reflection Coefficients Parametrization......Page 192
8.6.2 Riemannian Geometry of Toeplitz Covariance Matrices......Page 193
8.6.3 A Numerical Example......Page 195
8.6.4 Radar Simulations......Page 196
8.7 Conclusions......Page 199
References......Page 200
9.1 Historical Preamble......Page 202
9.2 Information Geometry Foundation......Page 204
9.3 Cartan\'s Symmetric Spaces......Page 208
9.4 Information Geometry Versus Optimal Transport Theory......Page 209
9.5 Cartan\'s Center of Mass and Emery\'s Exponential Barycenter......Page 212
9.6 Isobarycentric Flow......Page 216
9.7 Fourier Heat Equation Flow on 1D Graph  of HPD(n) Matrices......Page 217
9.8 Flow Preserving Covariance Matrix Toeplitz Structure......Page 218
9.9 Median by Fibration of Conformal Poincaré\'s Unit Disk......Page 220
9.10 Geometry of Space-Time Covariance Matrix......Page 222
9.11 Cartan-Siegel Homogeneous Domains: Siegel Disk......Page 223
9.12 Mostow/Berger\'s Fibration of Siegel Disk......Page 226
9.13 Hua Kernel for Cartan-Siegel Domains, Berezin Quantization and Geometric Lift......Page 229
9.14 Information Metric, Entropy Metric and Bergman Metric......Page 232
9.15 Complex Riccati Equation in Cartan-Siegel Domains......Page 234
9.16 Maslov Index for Shilov Boundary of Poincaré/Siegel Disks......Page 235
9.17 Radar Applications for Robust Ordered-Statistic Processing: OS-HDR-CFAR and OS-STAP......Page 236
9.17.1 Robust Doppler Processing: OS-HDR-CFAR......Page 237
9.17.2 Robust Space-Time Processing: OS-STAP......Page 239
9.18 Miscellaneous: Shape Manifold......Page 246
9.19 From Fréchet-Median Barycentre in Poincaré\'s Disk to Douady-Earle Conformal Barycentre on Its Boundary......Page 247
9.20 Conclusion......Page 251
References......Page 253
10.1 Introduction......Page 259
10.2.1 SAR Signal Statistics......Page 260
10.2.3 Information Geometry......Page 262
10.3.2 Standard Wishart Classifier......Page 264
10.3.3 Information Geometry......Page 266
10.4 Results on Real Data......Page 267
10.4.1 Classification Results......Page 268
10.4.2 Polarimetric Interpretation......Page 272
10.5 Conclusion......Page 276
References......Page 277
11.1 Introduction......Page 279
11.2.1 Radar Time Series Model......Page 280
11.3 Geometry of Complex Circular Multivariate  Gaussian Model......Page 282
11.4.1 Reflection Coefficients Parametrization......Page 283
11.4.2 Kahler Metric on Reflection Coefficients......Page 284
11.5.1 Velocity Profiles of Wake Turbulence......Page 285
11.5.2 Wake Turbulence Detection Based on Doppler Entropy......Page 286
11.5.3 GPU Computation for Wake Vortex Doppler Processing......Page 289
11.6 Discussion and Conclusion......Page 290
References......Page 291
12.1 Introduction......Page 293
12.3 Subspace Learning Methods......Page 295
12.4.1 Background Modeling Using Principal Component Analysis......Page 297
12.4.2 Background Modeling Using Independent  Component Analysis......Page 307
12.5 Classification Using Linear Discrimination Analysis......Page 310
12.6 Object Tracking......Page 313
12.7 Conclusion......Page 316
References......Page 318
13.1 Introduction......Page 322
13.2.2 Hestenes\' Algorithm......Page 324
13.2.3 Golub-Kahan Algorithm......Page 325
13.2.4 Tridiagonalization and Symmetric QR iteration......Page 326
13.2.5 Tridiagonalization and Divide and Conquer Algorithm......Page 328
13.3 Case Study: Real-time Face and Eye Tracking......Page 335
References......Page 338
14.1.1 Motivations......Page 340
14.1.2 Contributions......Page 341
14.1.3 Organization......Page 342
14.2.1 Non-Negative Matrix Factorization......Page 343
14.2.2 Applications to the Detection of Overlapping  Sound Events......Page 345
14.3.1 Template Learning......Page 346
14.3.2 Audio Stream Decomposition......Page 347
14.4.2 Non-Negative Matrix Factorization and Sparsity......Page 348
14.4.3 Problem Formulation and Multiplicative Update......Page 349
14.5.1 Definition and Properties of the Beta-Divergence......Page 352
14.5.2 Non-Negative Matrix Factorization and  the Beta-Divergence......Page 353
14.5.3 Problem Formulation and Multiplicative Update......Page 355
14.6 Evaluation and Results......Page 356
14.6.1 Polyphonic Music Transcription......Page 357
14.6.2 Drum Transcription......Page 359
14.6.3 Environmental Sound Detection......Page 362
14.7 Conclusion......Page 365
References......Page 367
15.1 Introduction......Page 371
15.2 Bregman Matrix Divergences......Page 373
15.3 Mean-Sivergence: A Generalization of Markowitz\' Mean-Variance Model......Page 378
15.4 On-line Learning in the Mean-Divergence Model......Page 383
15.5 Experiments on Learning in the Mean-Divergence Model......Page 390
15.6 Discussion......Page 392
15.7 Conclusion......Page 398
References......Page 399
16.1 Introduction......Page 401
16.2.1 Definition and Examples......Page 402
16.2.2 Dual Parametrization......Page 403
16.2.3 Bregman Divergences......Page 404
16.2.4 Bregman Centroids......Page 405
16.3.1 Statistical Mixtures......Page 406
16.3.2 Getting Mixtures......Page 407
16.4.1 Bregman Hard Clustering......Page 408
16.4.2 Kullback--Leibler Centroids as Geometric Projections......Page 409
16.4.3 Model Hard Clustering......Page 412
16.5.1 Presentation......Page 414
16.5.3 An Example with a Gaussian Mixture Model......Page 415
16.5.4 Examples with Other Exponential Families......Page 418
16.6.1 Experiments on Images......Page 419
16.6.2 Prediction of 3D Structures of RNA Molecules......Page 421
References......Page 423
17.1 Introduction......Page 425
17.2 Bayesian Filtering on Euclidean Spaces......Page 426
17.2.1 Sequential Monte Carlo......Page 427
17.3 Differential Geometry Tools......Page 428
17.4.1 General Scheme......Page 437
17.4.2 Point Estimates......Page 438
17.5 Application to Tracking with Unknown Time-Varying State Covariance......Page 440
17.5.1 Space of Positive Definite Matrices......Page 441
17.5.2 Particle Filter Implementation......Page 443
17.5.3 Simulation Results......Page 444
17.6 Conclusion......Page 446
References......Page 447
Index......Page 448