The Mathematical Foundations of Manifold Learning

Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical perspective on manifold learning, delving into the intersection of kernel learning, spectral graph theory, and differential geometry. Emphasis is placed on the remarkable interplay between graphs and manifolds, which forms the foundation for the widely-used technique of manifold regularization. This work is written to be accessible to a broad mathematical audience, including machine learning researchers and practitioners interested in understanding the theorems underlying popular manifold learning algorithms and dimensionality reduction techniques.