Approximating Local Homology from Samples

Abstract

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult to construct in higher dimensions. We show that the persistence diagrams used for estimating local homology can be approximated using families of Vietoris-Rips complexes, whose simpler construction are robust in any dimension. To the best of our knowledge, our results, for the first time make applications based on local homology, such as stratification learning, feasible in high dimensions. IntroductionAdvances in scientific and computational experiments have improved our ability to gather large collections of data points in high-dimensional spaces. One aspect in topological data analysis is to infer the topological structure of a space given a point cloud sample. We often assume the space has manifold structure, however, more interesting cases arise when we relax our assumptions to include spaces that contains singularities and mixed dimensionality, such as stratified spaces for example.Stratified spaces can be decomposed into manifold pieces that are glued together in some uniform way. An important tool in studying these spaces is the study of the neighborhoods surrounding singularities, where manifolds of different dimensionality intersect. We focus on sampling conditions for such neighborhoods, which allow us to begin examining how successful certain reconstruction techniques are with respect to the geometric properties of the underlying shape. Our goal is to infer sampling conditions for recovering local structures of stratified spaces, in particular, the local homology groups, from a possibly noisy sampled point set. Stratification learning. In stratification learning (or mixed manifold learning), a point cloud is assumed to be sampled from a mixture of (possibly intersecting) manifolds. The objective is to recover the different

Dimension Detection with Local Homology

“…Finally, we remark that similar to the recent work in [27], our computation of local homology uses the Rips complex, which is much easier to construct than the ambient Delaunay triangulation as was originally required in [2]. Different from [27], we aim to compute H(M, M − z) exactly for the special case when M is a manifold, while the work in [27] approximates the multiscale representations of local homology (the persistence diagram of certain filtration) for more general compact sets. We also note that, unlike [27] our algorithm operates with Rips complexes that span vertices within a local neighborhood, thus saving computations.…”

“…This line of work was further developed in [3] where the so-called local homology transfer was proposed to cluster points from different strata. In a recent paper [27], Skraba and Wang proposed to approximate the multi-scale representations of local homology using families of Rips complexes. Rips com-plexes are more suitable than the Delaunay triangulations for points sampled from low dimensional compact sets embedded in high dimensional space and have attracted much attention in topology inference [1,7,27].…”

Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives

“…9. We plan to build on the results of [5,28], and extend the sheaf-theoretic stratification learning perspective described in this paper to the study of stratifications of point cloud data using persistent local homology.…”

“…Statistical approaches that rely on inferences of mixture models and local dimension estimation require strict geometric assumptions such as linearity [16,19,29], and may not handle general scenarios with complex singularities. Recently, approaches from topological data analysis [3,5,28], which rely heavily on ingredients from computational [11] and intersection homology [2,4,12], are gaining momentum in stratification learning.…”

Another look at recovering local homology from samples of stratified sets

“…These simplicial complexes have nice theoretical properties, but are computationally efficient only in low dimensions. The result in [29] focused on an efficient approximation of a multi-scale representation of local homology using Vietoris-Rips complexes. However, the authors of the latter paper do not address the question of when such an approximation captures the true local homology.…”