Another look at recovering local homology from samples of stratified sets
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Another look at recovering local homology from samples of stratified sets Yuriy Mileyko1 Received: 22 November 2018 / Accepted: 3 November 2020 © Springer Nature Switzerland AG 2020
Abstract Recovering homological features of spaces from samples has become one of the central themes of topological data analysis, leading to many successful applications. Many of the results in this area focus on global homological features of a subset of a Euclidean space. In this case, homology recovery predicates on imposing well understood geometric conditions on the underlying set. Typically, these conditions guarantee that small enough neighborhoods of the set have the same homology as the set itself. Existing work on recovering local homological features of a space from samples employs similar conditions locally. However, such local geometric conditions may vary from point to point and can potentially degenerate. For instance, the size of local homology preserving neighborhoods across all points of interest may not be bounded away from zero. In this paper, we introduce more general and robust conditions for local homology recovery and show that tame homology stratified sets, including Whitney stratified sets, satisfy these conditions away from strata boundaries, thus obtaining control over the regions where local homology recovery may not be feasible. Moreover, we show that true local homology of such sets can be computed from good enough samples using Vietoris–Rips complexes. Keywords Topological data analysis · Stratified sets · Local homology · Vietoris–Rips complexes Mathematics Subject Classification 55U05 · 55U10 · 55N99 · 57N40 · 57N80
1 Introduction Estimating topological features of a space from samples is one of the central topics in topological data analysis (TDA), which is a new field that has been steadily gaining
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Yuriy Mileyko [email protected] Department of Mathematics, University of Hawai‘i at M¯anoa, Honolulu, HI, USA
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popularity due to a series of successful applications (see e.g. Ghrist 2008; Carlsson et al. 2008; de Silva and Ghrist 2007; Chan et al. 2013; Horak and Maleti´c 2009). The importance of such estimates stems from the fact that they provide us with a better insight into the process underlying the data, and can potentially help us select a better class of generative models. Much of the work within TDA focuses on developing and performing theoretical analyses of various methods for summarizing global homological properties of data sets. In particular, by imposing well understood geometric conditions on the underlying space, several guarantees for recovery of correct homology from sufficiently dense samples have been obtained (e.g. Niyogi et al. 2008, 2011; Cohen-Steiner et al. 2007; Chazal and Oudot 2008). Of course, one can easily make an argument that recovering global homological information may not be enough. Indeed, a space having the shape of the letter X is contractible, and thus has trivial homology, but the presence of a singular point may be extremely important.
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