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

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.

Introduction
Hà Quang Minh and Vittorio Murino

Bayesian Statistical Shape Analysis on the Manifold of Diffeomorphisms
Miaomiao Zhang and P. Thomas Fletcher

Sampling Constrained Probability Distributions using Spherical Augmentation
Shiwei Lan and Babak Shahbaba

Geometric Optimization in Machine Learning
Suvrit Sra and Reshad Hosseini

Positive Definite Matrices: Data Representation and Applications to Computer Vision
Anoop Cherian and Suvrit Sra

From Covariance Matrices to Covariance Operators: Data Representation from Finite to Infinite-Dimensional Settings
Hà Quang Minh and Vittorio Murino

Dictionary Learning on Grassmann Manifolds
Mehrtash Harandi, Richard Hartley, Mathieu Salzmann, and Jochen Trumpf

Regression on Lie Groups and its Application to Affine Motion Tracking
Fatih Porikli

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.