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Table of Contents
Preface; Overview and Goals; Acknowledgments; Contents; Contributors; Introduction; Themes of the Volume; Organization of the Volume; 1 Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms; 1.1 Introduction; 1.2 Mathematical Background; 1.2.1 Space of Diffeomorphisms; 1.2.2 Metrics on Diffeomorphisms; 1.2.3 Diffeomorphic Atlas Building with LDDMM; 1.3 A Bayesian Model for Atlas Building; 1.4 Estimation of Model Parameters; 1.4.1 Hamiltonian Monte Carlo (HMC) Sampling; 1.4.2 The Maximization Step; 1.5 Bayesian Principal Geodesic Analysis; 1.5.1 Probability Model
1.5.2 Inference1.6 Results; References; 2 Sampling Constrained Probability Distributions Using Spherical Augmentation; 2.1 Introduction; 2.2 Preliminaries; 2.2.1 Hamiltonian Monte Carlo; 2.2.2 Lagrangian Monte Carlo; 2.3 Spherical Augmentation; 2.3.1 Ball Type Constraints; 2.3.2 Box-Type Constraints; 2.3.3 General q-Norm Constraints; 2.3.4 Functional Constraints; 2.4 Monte Carlo with Spherical Augmentation; 2.4.1 Common Settings; 2.4.2 Spherical Hamiltonian Monte Carlo; 2.4.3 Spherical LMC on Probability Simplex; 2.5 Experimental Results; 2.5.1 Truncated Multivariate Gaussian
2.5.2 Bayesian Lasso2.5.3 Bridge Regression; 2.5.4 Reconstruction of Quantized Stationary Gaussian Process; 2.5.5 Latent Dirichlet Allocation on Wikipedia Corpus; 2.6 Discussion; References; 3 Geometric Optimization in Machine Learning; 3.1 Introduction; 3.2 Manifolds and Geodesic Convexity; 3.3 Beyond g-Convexity: Thompson Nonexpansivity; 3.3.1 Why Thompson Nonexpansivity?; 3.4 Manifold Optimization; 3.5 Applications; 3.5.1 Gaussian Mixture Models; 3.5.2 MLE for Elliptically Contoured Distributions; 3.5.3 Other Applications; References
4 Positive Definite Matrices: Data Representation and Applications to Computer Vision4.1 Introduction; 4.1.1 Covariance Descriptors and Example Applications; 4.1.2 Geometry of SPD Matrices; 4.2 Application to Sparse Coding and Dictionary Learning; 4.2.1 Dictionary Learning with SPD Atoms; 4.2.2 Riemannian Dictionary Learning and Sparse Coding; 4.3 Applications of Sparse Coding; 4.3.1 Nearest Neighbors on Covariance Descriptors; 4.3.2 GDL Experiments; 4.3.3 Riemannian Dictionary Learning Experiments; 4.3.4 GDL Versus Riemannian Sparse Coding; 4.4 Conclusion and Future Work; References
5 From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings5.1 Introduction; 5.2 Covariance Matrices for Data Representation; 5.3 Infinite-Dimensional Covariance Operators; 5.3.1 Positive Definite Kernels, Reproducing Kernel Hilbert Spaces, and Feature Maps; 5.3.2 Covariance Operators in RKHS and Data Representation; 5.4 Distances Between RKHS Covariance Operators; 5.4.1 Hilbert
Schmidt Distance; 5.4.2 Riemannian Distances Between Covariance Operators; 5.4.3 The Affine-Invariant Distance
1.5.2 Inference1.6 Results; References; 2 Sampling Constrained Probability Distributions Using Spherical Augmentation; 2.1 Introduction; 2.2 Preliminaries; 2.2.1 Hamiltonian Monte Carlo; 2.2.2 Lagrangian Monte Carlo; 2.3 Spherical Augmentation; 2.3.1 Ball Type Constraints; 2.3.2 Box-Type Constraints; 2.3.3 General q-Norm Constraints; 2.3.4 Functional Constraints; 2.4 Monte Carlo with Spherical Augmentation; 2.4.1 Common Settings; 2.4.2 Spherical Hamiltonian Monte Carlo; 2.4.3 Spherical LMC on Probability Simplex; 2.5 Experimental Results; 2.5.1 Truncated Multivariate Gaussian
2.5.2 Bayesian Lasso2.5.3 Bridge Regression; 2.5.4 Reconstruction of Quantized Stationary Gaussian Process; 2.5.5 Latent Dirichlet Allocation on Wikipedia Corpus; 2.6 Discussion; References; 3 Geometric Optimization in Machine Learning; 3.1 Introduction; 3.2 Manifolds and Geodesic Convexity; 3.3 Beyond g-Convexity: Thompson Nonexpansivity; 3.3.1 Why Thompson Nonexpansivity?; 3.4 Manifold Optimization; 3.5 Applications; 3.5.1 Gaussian Mixture Models; 3.5.2 MLE for Elliptically Contoured Distributions; 3.5.3 Other Applications; References
4 Positive Definite Matrices: Data Representation and Applications to Computer Vision4.1 Introduction; 4.1.1 Covariance Descriptors and Example Applications; 4.1.2 Geometry of SPD Matrices; 4.2 Application to Sparse Coding and Dictionary Learning; 4.2.1 Dictionary Learning with SPD Atoms; 4.2.2 Riemannian Dictionary Learning and Sparse Coding; 4.3 Applications of Sparse Coding; 4.3.1 Nearest Neighbors on Covariance Descriptors; 4.3.2 GDL Experiments; 4.3.3 Riemannian Dictionary Learning Experiments; 4.3.4 GDL Versus Riemannian Sparse Coding; 4.4 Conclusion and Future Work; References
5 From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings5.1 Introduction; 5.2 Covariance Matrices for Data Representation; 5.3 Infinite-Dimensional Covariance Operators; 5.3.1 Positive Definite Kernels, Reproducing Kernel Hilbert Spaces, and Feature Maps; 5.3.2 Covariance Operators in RKHS and Data Representation; 5.4 Distances Between RKHS Covariance Operators; 5.4.1 Hilbert
Schmidt Distance; 5.4.2 Riemannian Distances Between Covariance Operators; 5.4.3 The Affine-Invariant Distance