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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.
List of contents
Introduction.- Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms.- Sampling Constrained Probability Distributions using Spherical Augmentation.- Geometric Optimization in Machine Learning.- Positive Definite Matrices: Data Representation and Applications to Computer Vision.- From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings.- Dictionary Learning on Grassmann Manifolds.- Regression on Lie Groups and its Application to Affine Motion Tracking.- An Elastic Riemannian Framework for Shape Analysis of Curves and Tree-Like Structures.
About the author
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.
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.
Additional text
“The book under review consists of eight chapters, each introducing techniques for solving problems on manifolds and illustrating these with examples. … reading this book would add to my collection of tools for working with data on manifolds and expose me to new problems treatable by these tools. … In each case an effort has been made to provide enough of the underlying theory supporting the techniques, with explicit references where the interested reader can go for further details.” (Tim Zajic, IAPR Newsletter, Vol. 39 (3), July, 2017)
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"The book under review consists of eight chapters, each introducing techniques for solving problems on manifolds and illustrating these with examples. ... reading this book would add to my collection of tools for working with data on manifolds and expose me to new problems treatable by these tools. ... In each case an effort has been made to provide enough of the underlying theory supporting the techniques, with explicit references where the interested reader can go for further details." (Tim Zajic, IAPR Newsletter, Vol. 39 (3), July, 2017)
