Elements of Dimensionality Reduction and Manifold Learning

Dimensionality reduction, also known as manifold learning, is an area of machine learning used for extracting informative features from data, for better representation of data or separation between classes. This book presents a cohesive review of linear and nonlinear dimensionality reduction and manifold learning. Three main aspects of dimensionality reduction are covered -- spectral dimensionality reduction, probabilistic dimensionality reduction, and neural network-based dimensionality reduction, which have geometric, probabilistic, and information-theoretic points of view to dimensionality reduction, respectively. This book delves into basic concepts and recent developments in the field of dimensionality reduction and manifold learning, providing the reader with a comprehensive understanding. The necessary background and preliminaries, on linear algebra, optimization, and kernels, are also explained in the book to ensure a comprehensive understanding of the algorithms.The tools this book introduces are applied to various applications involving feature extraction, image processing, computer vision, and signal processing. This book is applicable to a wide audience who would like to acquire a deep understanding of the various ways to extract, transform, and understand the structure of data. The intended audiences are academics, students, and industry professionals. Academic researchers and students can use this book as a textbook for machine learning and dimensionality reduction. Data scientists, machine learning scientists, computer vision scientists, computer scientists, statisticians, and mathematicians in the fields of applied mathematics, statistical learning, Riemannian manifolds, subspace analysis, linear algebra, and optimization can use this book as a reference book for both technical and applied concepts. Industry professionals, including applied engineers, data engineers, and engineers in various fields of science dealing with machine learning, can use this book as a guidebook for feature extraction from their data as the raw data in industry often requires pre-processing. Data feature extraction is useful in various fields of science including engineering, physics, chemistry, biometrics, biomedical signals and images, etc.This book is structured as a reference textbook, so that it can be used for advanced courses, as an in-depth supplementary resource or for researchers or practitioners who want to learn about dimensionality reduction and manifold learning. The book is grounded in theory, but provides thorough explanations and diverse examples to improve the readers comprehension of the advanced topics. Advanced methods are explained in a step-by-step manner so that readers of all levels can follow the reasoning and come to a deep understanding of the concepts. This book does not assume the reader has an advanced theoretical background in machine learning and provides necessary background, though an undergraduate-level background in linear algebra and calculus is recommended.