Simplicial Hausdorff Distance for Topological Data Analysis

TL;DR

This paper introduces a new metric called the simplicial Hausdorff distance that combines geometric and topological information of simplicial complexes for better analysis in topological data analysis.

Contribution

It proposes an extended Hausdorff distance integrating geometric proximity with topological features and discusses its computational complexity and properties.

Findings

Extended metric effectively captures geometric and topological relationships.

Computational complexity analysis of the proposed metric.

Discussion on monotonicity of measurement functions.

Abstract

Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between the elements of the complex. In addition to the combinatorial structure, simplicial complexes possess a geometric realization that provides a concrete way to visualize the complex and understand its geometric properties. This work presents an amended Hausdorff distance as an extended metric that integrates geometric proximity with the topological features of simplicial complexes. We also present a version of the simplicial Hausdorff metric for filtered complexes and show results on its computational complexity. In addition, we discuss concerns about the monotonicity of the measurement functions involved in the setup of the simplicial complexes.