Maximum persistent Betti numbers of Čech complexes

Area of Science:

• Computational Topology
• Geometric Analysis
• Discrete Geometry

Background:

• Čech complexes are fundamental in topological data analysis for representing shape.
• Understanding the persistence of topological features (holes) is crucial for data interpretation.
• Previous work suggested bounds, but explicit proofs for persistent holes over intervals were lacking.

Purpose of the Study:

• To prove that the number of p-dimensional holes in a Čech complex of n points in R^d persisting over a constant length interval is linearly bounded by n.
• To provide an elementary and self-contained proof for this linear bound.
• To demonstrate the applicability of the bound to Alpha and Vietoris-Rips complexes.

Main Methods:

• A packing argument is employed.
• Čech complexes are related to snap complexes over a spatial partition.
• The proof relies on geometric and combinatorial constructions.

Main Results:

• A linear upper bound (constant times n) is established for the number of p-dimensional holes persisting from radius 1 to 1+ε.
• The bound holds for any fixed dimension p < d and ε > 0.
• The result is shown to apply to Alpha complexes and Vietoris-Rips complexes.

Conclusions:

• The number of persistent holes in Čech complexes, Alpha complexes, and Vietoris-Rips complexes over a fixed interval is shown to be linearly dependent on the number of data points.
• This provides a fundamental quantitative understanding of topological feature persistence.
• The elementary proof offers a new perspective without relying on advanced theories.