Offset hypersurfaces and persistent homology of algebraic varieties

Abstract

In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and algebraic optimization. Namely, we express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean Distance Degree of the starting variety, obtaining a new way to compute these degrees. Finally, we describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.

Sampling Real Algebraic Varieties for Topological Data Analysis

“…Future work will explore how to exploit the algebraic description of a variety in computing these quantities. Finally, it would be worthwhile to investigate the noise induced from sampling via homotopy continuation in the context of off-set varieties [36].…”

Learning algebraic varieties from samples

“…We discuss the work of Horobeţ and Weinstein in [32] which concerns metric properties of a given variety V ⊂ R n that are relevant for its true persistent homology. Here, the true persistent homology of V , at parameter value , refers to the homology of the -neighborhood of V .…”

“…It is shown in [32,Theorem 3.4] that the endpoints of bars in the true persistent homology of a variety V occur at numbers that are algebraic over Q. The proof relies on results in real algebraic geometry that characterize the family of fibers in a map of semialgebraic sets.…”

“…At present we do not know a good formula or a tight bound for the algebraic degrees of the barcode and the reach in terms of the invariants of the variety V . Deriving such formulas will require a further development and careful analysis of the offset discriminant that was introduced in [32]. We hope to return to this topic in the near future, as it can play a fundamental link between topology and algebraic geometry in the context of data science.…”

Universally consistent estimation of the reach

“…The reach has gained importance in the last two decades in the areas of statistics known as set estimation, manifold learning and persistent homology (see for instance Cuevas, Fraiman and Pateiro-López (2016), Arias-Castro et al (2020) and Horobeţ et al (2019), respectively). Given an unknown set M ⊂ R d (not necessarily convex), set estimation deals with the problem of the estimation of M from a random sample {X 1 , .…”