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This summary is machine-generated.

The number of persistent holes in Čech complexes of n points in Euclidean space is linear with respect to n. This finding bounds topological features persisting over a fixed interval, applicable to various complexes.

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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.