Algorithmic Advances in Riemannian Geometry and Applications (eBook)

Dr. Hà Quang Minh is a researcher in the Pattern Analysis and Computer Vision (PAVIS) group, at the Italian Institute of Technology (IIT), in Genoa, Italy. Dr. Vittorio Murino is a full professor at the University of Verona Department of Computer Science, and the Director of the PAVIS group at the IIT.

Preface 6
Overview and Goals 6
Acknowledgments 7
Contents 8
Contributors 9
Introduction 11
Themes of the Volume 11
Organization of the Volume 12
1 Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms 15
1.1 Introduction 15
1.2 Mathematical Background 17
1.2.1 Space of Diffeomorphisms 17
1.2.2 Metrics on Diffeomorphisms 18
1.2.3 Diffeomorphic Atlas Building with LDDMM 19
1.3 A Bayesian Model for Atlas Building 20
1.4 Estimation of Model Parameters 21
1.4.1 Hamiltonian Monte Carlo (HMC) Sampling 23
1.4.2 The Maximization Step 24
1.5 Bayesian Principal Geodesic Analysis 25
1.5.1 Probability Model 26
1.5.2 Inference 27
1.6 Results 29
References 35
2 Sampling Constrained Probability Distributions Using Spherical Augmentation 38
2.1 Introduction 38
2.2 Preliminaries 40
2.2.1 Hamiltonian Monte Carlo 40
2.2.2 Lagrangian Monte Carlo 41
2.3 Spherical Augmentation 42
2.3.1 Ball Type Constraints 42
2.3.2 Box-Type Constraints 43
2.3.3 General q-Norm Constraints 44
2.3.4 Functional Constraints 46
2.4 Monte Carlo with Spherical Augmentation 49
2.4.1 Common Settings 49
2.4.2 Spherical Hamiltonian Monte Carlo 50
2.4.3 Spherical LMC on Probability Simplex 56
2.5 Experimental Results 59
2.5.1 Truncated Multivariate Gaussian 60
2.5.2 Bayesian Lasso 61
2.5.3 Bridge Regression 64
2.5.4 Reconstruction of Quantized Stationary Gaussian Process 65
2.5.5 Latent Dirichlet Allocation on Wikipedia Corpus 67
2.6 Discussion 69
References 81
3 Geometric Optimization in Machine Learning 85
3.1 Introduction 85
3.2 Manifolds and Geodesic Convexity 86
3.3 Beyond g-Convexity: Thompson Nonexpansivity 88
3.3.1 Why Thompson Nonexpansivity? 89
3.4 Manifold Optimization 90
3.5 Applications 93
3.5.1 Gaussian Mixture Models 93
3.5.2 MLE for Elliptically Contoured Distributions 96
3.5.3 Other Applications 99
References 100
4 Positive Definite Matrices: Data Representation and Applications to Computer Vision 104
4.1 Introduction 104
4.1.1 Covariance Descriptors and Example Applications 106
4.1.2 Geometry of SPD Matrices 108
4.2 Application to Sparse Coding and Dictionary Learning 109
4.2.1 Dictionary Learning with SPD Atoms 109
4.2.2 Riemannian Dictionary Learning and Sparse Coding 112
4.3 Applications of Sparse Coding 116
4.3.1 Nearest Neighbors on Covariance Descriptors 116
4.3.2 GDL Experiments 117
4.3.3 Riemannian Dictionary Learning Experiments 118
4.3.4 GDL Versus Riemannian Sparse Coding 122
4.4 Conclusion and Future Work 123
References 123
5 From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings 126
5.1 Introduction 126
5.2 Covariance Matrices for Data Representation 129
5.3 Infinite-Dimensional Covariance Operators 133
5.3.1 Positive Definite Kernels, Reproducing Kernel Hilbert Spaces, and Feature Maps 133
5.3.2 Covariance Operators in RKHS and Data Representation 135
5.4 Distances Between RKHS Covariance Operators 136
5.4.1 Hilbert--Schmidt Distance 136
5.4.2 Riemannian Distances Between Covariance Operators 138
5.4.3 The Affine-Invariant Distance 142
5.5 Two-Layer Kernel Machines with RKHS Covariance Operators 145
5.5.1 The Interplay Between Positive Definite Kernels and Riemannian Manifolds 145
5.5.2 Two-Layer Kernel Machines 146
5.6 Experiments in Image Classification 146
5.7 Discussion, Conclusion, and Future Work 148
References 153
6 Dictionary Learning on Grassmann Manifolds 155
6.1 Introduction 155
6.2 Problem Statement 157
6.3 Background Theory 160
6.4 Dictionary Learning on Grassmannian 168
6.4.1 Weighted Karcher Mean 168
6.4.2 Dictionary Learning 172
6.5 Kernel Coding 173
6.5.1 Kernel-Based Riemannian Coding 176
6.5.2 Kernel Dictionary Learning 177
6.6 Experiments 179
References 180
7 Regression on Lie Groups and Its Application to Affine Motion Tracking 183
7.1 Introduction 183
7.2 Lie Group 184
7.3 Linear Regression on Matrix Lie Groups 185
7.4 Application to Affine Motion Tracking 187
7.4.1 Related Work 187
7.4.2 Tracking as a Regression Problem on Lie Group 189
References 194
8 An Elastic Riemannian Framework for Shape Analysis of Curves and Tree-Like Structures 196
8.1 Introduction 196
8.1.1 From Discrete to Continuous and Elastic 197
8.1.2 General Elastic Framework 198
8.2 Shape Analysis of Euclidean Curves 199
8.3 Shape Analysis of Trajectories in Hilbert Spaces 203
8.3.1 Elastic Comparison of Trajectories in mathbbL2([0,1],mathbbR) 203
8.3.2 Elastic Comparison of Trajectories in mathbbL2([0,1],mathbbR)/? 205
8.4 Elastic Shape Analysis of Axonal Trees 206
8.4.1 Representing Trees as Composite Trajectories 206
8.4.2 Shape Space of Axonal Trees and Geodesic Paths 209
8.4.3 Experimental Results 209
8.5 Karcher Mean of Tree-Like Structures 212
References 213
Index 215