The Mathematical Foundations of Manifold Learning

This is an edited version of my undergraduate thesis, submitted to the Harvard Mathematics Department in May 2020. It differs from the original thesis in one major respect, namely that this version omits the proofs of a number of theorems that are readily-available in other expositions. Whereas the original version reproduced these proofs in full, this version simply contains references to these proofs in other works. This thesis is built upon an extensive body of prior work in learning theory, graph theory, differential geometry, and manifold learning. In particular, I would like to thank Professors Lorenzo Rosasco and Tomaso Poggio for their lectures on statistical learning theory, Professor Daniel Spielman for his notes on spectral graph theory, Professor Yaiza Canzani for her notes on analysis on manifolds, and Professor Mikhail Belkin for his work on manifold learning. Finally, I wish to thank those people without whom I could never have written this thesis: my family, friends, and wonderful advisor Professor Arjun Manrai. Unlike the manifolds discussed herein, their support was truly boundless. I hope you enjoy and learn something from this thesis! If you have comments, corrections, or would like to contact me for anything else, feel free to email me.