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SUMMARY:Manifold Fitting: an Invitation to Data Science - Zhigang Yao (Nat
 ional University of Singapore)
DTSTART:20231124T140000Z
DTEND:20231124T150000Z
UID:TALK206023@talks.cam.ac.uk
CONTACT:Qingyuan Zhao
DESCRIPTION:While classical statistics has dealt with observations which a
 re real numbers or elements of a real vector space\, nowadays many statist
 ical problems of high interest in the sciences deal with the analysis of d
 ata which consist of more complex objects\, taking values in spaces which 
 are naturally not (Euclidean) vector spaces but which still feature some g
 eometric structure. The manifold fitting problem can go back to H. Whitney
 ’s work in the early 1930s (Whitney (1992))\, and finally has been answe
 red in recent years by C. Fefferman’s works (Fefferman\, 2006\, 2005). T
 he solution to the Whitney extension problem leads to new insights for dat
 a interpolation and inspires the formulation of the Geometric Whitney Prob
 lems (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 su
 bmanifold $\\widehat{M} \\subset \\mathbb{R}^D$ to approximate $Y$\, and h
 ow well can $\\widehat{M}$ estimate $Y$ in terms of distance and smoothnes
 s? To address these problems\, various mathematical approaches have been p
 roposed (see Fefferman et al. (2016\, 2018\, 2021b)). However\, many of th
 ese methods rely on restrictive assumptions\, making extending them to eff
 icient and workable algorithms challenging. As the manifold hypothesis (no
 n-Euclidean structure exploration) continues to be a foundational element 
 in statistics\, the manifold fitting problem\, merits further exploration 
 and discussion within the modern data science community. This talk will be
  partially based on recent works of Yao and Xia (2019) and Yao\, Su\, Li a
 nd Yau (2022).
LOCATION:MR12\, Centre for Mathematical Sciences
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