  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.
