Simplicial Hausdorff Distance for Topological Data Analysis

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.