Manifold Fitting: an Invitation to Machine Learning – a Mathematician’s view

Abstract: Natural datasets have intrinsic patterns, which can be summarized as the manifold distribution principle: the distribution of a class of data is close to a low-dimensional manifold. The manifold fitting problem can go back to the solution to the Whitney extension problem leading to new insights for data interpolation. Assume that we are given a set $Y\subseteq\mathbb{R}^D$. When can we construct a smooth d-dimensional submanifold $\widehat{M}\subseteq\mathbb{R}^D$ to approximate $Y$, and how well can $\widehat{M}$ estimate $Y$ in terms of distance and smoothness? However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in data science, the manifold fitting problem, merits further exploration and discussion within the modern data science community. The talk will be partially based on some recent works [4, 2, 3, 1] along with some on-going progress.

[1] Zhigang Yao, Bingjie Li, Yukun Lu, and Shing-Tung Yau. Single-cell analysis via manifold fitting: A new framework for RNA clustering and beyond, 2024.

[2] Zhigang Yao, Jiaji Su, Bingjie Li, and Shing-Tung Yau. Manifold fitting. arXiv preprint 2304.07680, 2023.

[3] Zhigang Yao, Jiaji Su, and Shing-Tung Yau. Manifold fitting with cycleGAN. Proceedings of the National Academy of Sciences of the United States of America, 121(5):e2311436121, 2023.

[4] Zhigang Yao and Yuqing Xia. Manifold fitting under unbounded noise. arXiv preprint 1909.10228, 2019.

machine learningalgebraic topologydifferential geometrygeometric topologymetric geometry

Audience: researchers in the topic

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Barcelona Mathematics and Machine Learning Colloquium Series

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