﻿ Manifold fitting: an invitation to statistics-清华丘成桐数学科学中心

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## Manifold fitting: an invitation to statistics

Abstract:

While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. The manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set \$Y \subset \mathbb{R}^D\$. When can we construct a smooth \$d\$-dimensional submanifold \$\widehat{M} \subset \mathbb{R}^D\$ to approximate \$Y\$, and how well can \$\widehat{M}\$ estimate \$Y\$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). 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 statistics, the manifold fitting problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on some recent works of Yao with his co-authors  along with some on-going progress . Relevant references:https://arxiv.org/abs/1909.10228 and https://arxiv.org/abs/2304.07680.

Yao Zhigang, 新加坡国立大学统计与数据科学系副教授兼终身教授。他现为哈佛大学数学科学与应用中心成员，哈佛大学统计系访问教授，清华大学YMSC访问教授，也曾作为特邀客座教授访问瑞士洛桑联邦理工大学（EPFL）等大学。 研究兴趣主要是复杂数据的统计推断。近年来专注于非欧式统计(Non-Euclidean Statistics)和低维流形学习。Yao在与Prof Shing-Tung Yau合作和帮助下，致力于推动几何和统计的交互这个全新的领域（https://cmsa.fas.harvard.edu/event/geometry-and-statistics/）。Yao也是即将在中国召开的第一届几何和统计交互的研讨会的倡导者 （https://zhigang-yao.github.io/bimsa-satellite/）。 近年来，Yao与其合作者提出在黎曼流形上重新定义传统PCA的principal flow/sub-manifold以及principal boundary等方法和理论，以及全空间下新的manifold learning方法和理论。这些方法通过考虑数据本身的非欧结构，旨在解决传统统计方法和理论中的缺陷。个人网页 https://zhigang-yao.github.io/