Algorithmic Advances in Riemannian Geometry and Applications
For Machine Learning, Computer Vision, Statistics, and Optimization

Editors: Hà Quang Minh, Vittorio Murino



Showcases Riemannian geometry as a foundational mathematical framework for solving many problems in machine learning, statistics, optimization, computer vision, and related fields

Describes comprehensively the state-of-the-art theory and algorithms in the Riemannian framework along with their concrete practical applications

Written by leading experts in statistics, machine learning, optimization, pattern recognition, and computer vision

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.

Table of contents

Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms

Sampling Constrained Probability Distributions Using Spherical Augmentation

Geometric Optimization in Machine Learning

Positive Definite Matrices:

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