Algorithmic advances in Riemannian geometry and applications : for machine learning, computer vision, statistics, and optimization

Description

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. �Read more... Abstract: 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. �Read more... Front Matter ....Pages i-xiv Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms (Miaomiao Zhang, P. Thomas Fletcher)....Pages 1-23 Sampling Constrained Probability Distributions Using Spherical Augmentation (Shiwei Lan, Babak Shahbaba)....Pages 25-71 Geometric Optimization in Machine Learning (Suvrit Sra, Reshad Hosseini)....Pages 73-91 Positive Definite Matrices: DataRepresentation and Applications to Computer Vision (Anoop Cherian, Suvrit Sra)....Pages 93-114 From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings (Hà Quang Minh, Vittorio Murino)....Pages 115-143 Dictionary Learning on Grassmann Manifolds (Mehrtash Harandi, Richard Hartley, Mathieu Salzmann, Jochen Trumpf)....Pages 145-172 Regression on Lie Groups and Its Application to Affine Motion Tracking (Fatih Porikli)....Pages 173-185 An Elastic Riemannian Framework for Shape Analysis of Curves and Tree-Like Structures (Adam Duncan, Zhengwu Zhang, Anuj Srivastava)....Pages 187-205 Back Matter ....Pages 207-208