| Description |
1 online resource : illustrations |
| Series |
Advances in computer vision and pattern recognition |
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Advances in computer vision and pattern recognition.
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| 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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
This book presents a selection of the most recent algorithmic advances in Riemannian geometry in the context of machine learning, statistics, optimization, computer vision, and related fields. The unifying theme of the different chapters in the book is the exploitation of the geometry of data using the mathematical machinery of Riemannian geometry. As demonstrated by all the chapters in the book, when the data is intrinsically non-Euclidean, the utilization of this geometrical information can lead to better algorithms that can capture more accurately the structures inherent in the data, leading ultimately to better empirical performance. This book is not intended to be an encyclopedic compilation of the applications of Riemannian geometry. Instead, it focuses on several important research directions that are currently actively pursued by researchers in the field. These include statistical modeling and analysis on manifolds, optimization on manifolds, Riemannian manifolds and kernel methods, and dictionary learning and sparse coding on manifolds. Examples of applications include novel algorithms for Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis, image classification, action recognition, and motion tracking |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Geometry, Riemannian.
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Riemannian manifolds.
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Machine learning.
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Computer vision.
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Statistics.
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optimization
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statistics.
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Artificial intelligence.
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Mathematical & statistical software.
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Mathematical modelling.
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Maths for computer scientists.
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Pattern recognition.
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MATHEMATICS -- Geometry -- General.
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Computer vision
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Geometry, Riemannian
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Machine learning
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Riemannian manifolds
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Statistics
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| Form |
Electronic book
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| Author |
Minh, Ha Quang, editor
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Murino, Vittorio, editor.
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| ISBN |
9783319450261 |
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3319450263 |
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