Manifold Fitting

Date: Thursday 23 November 2023, 2.00PM
Location: University of Kent, Canterbury
Registry Meeting Room G03, Registry, University of Kent
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Abstract of the talk:
This 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 recent works of Yao and Xia (2019) and Yao, Su, Li and Yau (2022).

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Dr Zhigang Yao, Department of Statistics and Data Science, National University of Singapore, 21 Lower Kent Ridge Road, Singapore 117546

Dr Zhigang Yao's personal webpage

Dr Zhigang Yao is a tenured associate professor in the Department of Statistics and Data Science at the National University of Singapore. He has been a visiting faculty at the Center for Mathematical Sciences and Applications at Harvard University since 2022. He holds a visiting professorship at YMSC at Tsinghua University.  His primary research interests lie in statistical inference for complex data. In recent years, his focus has shifted towards non-Euclidean statistics and low-dimensional manifold learning.
Yao is committed to promoting the new field of interaction between geometry and statistics. In recent years, Yao and his collaborators have proposed methods and theories that redefine traditional PCA  on Riemannian manifolds including principal flows/sub-manifolds and principal boundaries, as well as new manifold learning theories. These methods aim to address deficiencies in traditional statistical methods and theories by taking into account the geometry of the data.
Contact: Dr Peng Liu