Shing-Tung Yau

will talk about 'Manifold Fitting, an Invitation to Machine Learning – a Mathematician’s view'

page thumbnail

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 YD. When can we construct a smooth d-dimensional submanifold MD 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.

Click here to convert to your local time: 18th April 14:00 CET

Click here to download the slides.