Shing Tung Yau - Manifold Fitting: an Invitation to Machine Learning..., b=M2L 2024 - UAB

Barcelona Mathematics and Machine Learning (b=M2L) Colloquium Series

The Barcelona Mathematics and Machine Learning (b=M2L) Colloquium Series aims to bring to a general audience of mathematicians, computer scientists and interested students the interactions between mathematics and machine learning, as well as their latest developments. What are the mathematics behind machine learning? What can machine learning do for mathematicians?

Manifold Fitting, an Invitation to Machine Learning – a Mathematician’s view, 18th April 2024, 14:00 CET
Shing-Tung Yau - Tsinghua University, Beijing

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⊆ℝ^D. When can we construct a smooth d-dimensional submanifold M⊆ℝ^D to approximate Y , and how well can 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.

This activity is organized by the Department of Mathematics at the Universitat Autònoma de Barcelona with the goal of gathering together with the Barcelona universities and research centres with the global community.

[gerryg]

Been reading these papers. Great to have someone present manifold learning-related results altogether and publish them online. Thank you Professor Yau.