Regression and Fitting on Manifold-valued Data

This book introduces in a constructive manner a general framework for regression and fitting methods for many applications and tasks involving data on manifolds. The methodology has important and varied applications in machine learning, medicine, robotics, biology, computer vision, human biometrics, nanomanufacturing, signal processing, and image analysis, etc.

The first chapter gives motivation examples, a wide range of applications, raised challenges, raised challenges, and some concerns. The second chapter gives a comprehensive exploration and step-by-step illustrations for Euclidean cases. Another dedicated chapter covers the geometric tools needed for each manifold and provides expressions and key notions for any application for manifold-valued data.

All loss functions and optimization methods are given as algorithms and can be easily implemented. In particular, many popular manifolds are considered with derived and specific formulations. The same philosophy is used in all chapters and all novelties are illustrated with intuitive examples. Additionally, each chapter includes simulations and experiments on real-world problems for understanding and potential extensions for a wide range of applications.

Covers the topic in a step-by-step manner Includes simulations for understanding and potential experiments for a wide range of applications Covers optimization on most used manifolds in machine learning

Autorentext

Ines ADOUANI received her PhD in complex analysis and Finsler geometry from the University of Pierre and Marie Curie, France, in 2015. Since then, she has been serving as an Assistant Professor at the Institute of Applied Sciences and Technology of Sousse, Tunisia. Additionally, from 2020 to 2021, she held a position as an Assistant Professor at King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. Her main research interests encompass complex geometry (including Finsler and Kähler geometry), optimization on Riemannian manifolds, regression, and fitting on Riemannian manifolds, as well as their applications to computer vision and medical imaging problems.

Chafik SAMIR received his PhD on learning and analysis of shapes and patterns in 2007 at the University of Lille, France. After spending two years as a postdoc working on manifolds and related applications at UCL, he joined UCA in 2009. His main research interests are machine learning for manifold-valued data, such as functional and medical observations, optimization of loss functions, statistical shape analysis, spatio-temporal patterns and fusion, regression and fitting on Riemannian manifolds, and their applications to real-world problems.

Inhalt

Introduction.- Spline Interpolation and Fitting in R𝒏.- Spline Interpolation on the Sphere S𝒏.- Spline Interpolation on the Special Orthogonal Group 𝑺𝑶(𝒏).- Spline Interpolation on Stiefel and Grassmann manifolds.- Spline Interpolation on the Manifold of Probability Measures.- Spline Interpolation on the Manifold of Probability Density Functions.- Spline Interpolation on Shape Space.- Spline Interpolation on Other Riemannian Manifolds.