Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningMachine Learning: Learning with ManifoldsData Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
Simplicial Hausdorff Distance for Topological Data Analysis
🤖 AI Summary
This paper addresses the challenge of jointly modeling geometric proximity and topological features in topological data analysis (TDA). To this end, we propose a corrected simplicial Hausdorff distance for point-cloud-generated simplicial complexes. This is the first systematic extension of the classical Hausdorff metric to the category of simplicial complexes, rigorously satisfying the axioms of an extended metric while being simultaneously sensitive to both geometric structure and topological connectivity. We further construct a computable version of this distance on filtered complexes, analyze its time complexity, and establish the necessity and constraints imposed by monotonicity of the underlying filtration function on the distance definition. Experiments demonstrate that the proposed metric robustly captures structural evolution across filtration scales, offering a new tool for simplicial complex comparison, stability analysis, and machine learning embedding in TDA.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Develops a new Hausdorff distance metric
Integrates geometric and topological features
Analyzes computational complexity and monotonicity
Innovation
Methods, ideas, or system contributions that make the work stand out.
Extended Hausdorff distance metric
Simplicial complexes geometric proximity
Filtered complexes computational complexity
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