Benyamin Ghojogh - 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. With the explosion of interest and advances in machine learning, there has been a corresponding increased need for educational and reference books to explain various aspects of machine learning. However, there has not been a comprehensive text tackling the various methods in dimensionality reduction, manifold learning, and feature extraction that integrate with modern machine learning theory and practice.

This book presents a cohesive review of linear and nonlinear dimensionality reduction and manifold learning. Three main aspects of dimensionality reduction are coveredspectral dimensionality reduction, probabilistic dimensionality reduction, and neural network-based dimensionality reductionwhich 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 highlighted to ensure a comprehensive understanding of the algorithms. The tools introduced in this book can be applied to various applications involving feature extraction, image processing, computer vision, and signal processing.

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 an advanced theoretical background in machine learning and provides the necessary background, although an undergraduate-level background in linear algebra and calculus is recommended.

The book can be a resource for instructors teaching advanced undergraduate or graduate level courses in engineering, computer science, mathematics, and science. There are various corresponding courses that can use this book as their textbook. Some of these courses are machine learning, data science, artificial intelligence, unsupervised machine learning, data clustering, dimensionality reduction, manifold learning, manifold embedding, feature extraction, feature embedding, feature engineering, data visualization, etc. This book can also be considered a reference book for dimensionality reduction and manifold learning. It can also be seen as a history book for dimensionality reduction as a particular field of machine learning, as it presents the development of these concepts from inception.

This book is divided into four main sections: preliminaries and background concepts, spectral or geographic methods, probabilistic methods, and neural network-based methods.