SIAM Conference on Applied Linear Algebra

2024 SIAM Conference on Applied Linear Algebra

Part of MS45 Exploiting Low-dimensional Structures in Data Science - Part II of II

Manifold Fitting: An Invitation to Data Science

Abstract. 

This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)). 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 recent works of Yao and Xia (2019) and Yao, Su, Li and Yau (2022) and Yao, Su and Yau (2023)