# The Mathematical Foundations of Manifold Learning

arXiv: https://arxiv.org/abs/2011.01307 · HTML: https://mlsanity.com/paper/2011.01307/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [Deep Spectral Methods: A Surprisingly Strong Baseline for Unsupervised Semantic Segmentation and Localization](https://mlsanity.com/paper/2205.07839/) (2022)
- [ADEPT: A DEbiasing PrompT Framework](https://mlsanity.com/paper/2211.05414/) (2022)
- [Position: Cracking the Code of Cascading Disparity Towards Marginalized Communities](https://mlsanity.com/paper/2406.01757/) (2024)
- [Error Slice Discovery via Manifold Compactness](https://mlsanity.com/paper/2501.19032/) (2025)
- [Datacube segmentation via Deep Spectral Clustering](https://mlsanity.com/paper/2401.17695/) (2024)
- [Dissecting embedding method: learning higher-order structures from data](https://mlsanity.com/paper/2410.10917/) (2024)
- [FedMP: Tackling Medical Feature Heterogeneity in Federated Learning from a Manifold Perspective](https://mlsanity.com/paper/2508.09174/) (2025)

## Abstract (arXiv)

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.