2019/06/18 by Henry‐Louis de Kergorlay, de Kergorlay, Henry-Louis, Ulrike Tillmann +3 · 1 citation
Computer Science · Mathematics · Medicine · #05E45 #53B21 #55U10 #60B99 #60D05 #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1906.07626
openalex publication_date 2019/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random Čech-complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the k-th homology of M with high probability, and a lower threshold below which it almost certainly does not. These thresholds are close together in the sense that they have the same leading term. Here k is positive and strictly less than the dimension d of the manifold. This extends work of Bobrowski and Weinberger in [BW17] and Bobrowski and Oliveira [BO19] who establish similar formulas when M is a torus and, more generally, is closed and has no boundary. We note that the cases with and without boundary lead to different answers: The corresponding common leading terms for the upper and lower thresholds differ being log (n) when M is closed and (2-2/d)log (n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle, which we call a Θ-like-cycle, which occur close to the boundary and establish that the first order term of the lower threshold is (2-2/d)log (n).