Topological inference of manifolds with boundary

Yuan Wang, Bei Wang*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction – how to assemble discrete samples into global structures, and inference – how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have shown that by constructing an offset of the samples using a suitable offset parameter could provide reconstructions that preserve homotopy types therefore homology for densely sampled smooth submanifolds of Euclidean space without boundary. Chazal et al. (2009) and Attali et al. (2013) have introduced a parameterized set of sampling conditions that extend the results of Niyogi et al. to a large class of compact subsets of Euclidean space. Our work tackles data problems that fill a gap between the work of Niyogi et al. and Chazal et al. In particular, we give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories. We also give stronger results that relate topological equivalence between the offset and the manifold as a deformation retract.

Original languageEnglish (US)
Article number101606
JournalComputational Geometry: Theory and Applications
Volume88
DOIs
StatePublished - Jun 2020

Keywords

  • Computational topology
  • Sampling
  • Topological data analysis
  • Topological inference

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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