Maximum persistent Betti numbers of Čech complexes.

Abstract

This note proves that only a linear number of holes in a Čech complex of <i>n</i> points in Rd can persist over an interval of constant length. Specifically, for any fixed dimension p<d and fixed   ε>0 , the number of <i>p</i>-dimensional holes in the Čech complex at radius 1 that persist to radius 1+ε is bounded above by a constant times <i>n</i>, where <i>n</i> is the number of points. The proof uses a packing argument supported by relating the Čech complexes with corresponding snap complexes over the cells in a partition of space. The argument is self-contained and elementary, relying on geometric and combinatorial constructions rather than on the existing theory of sparse approximations or interleavings. The bound also applies to Alpha complexes and Vietoris-Rips complexes. While our result can be inferred from prior work on sparse filtrations, to our knowledge, no explicit statement or direct proof of this bound appears in the literature.