TOPOGRAPH: Topology-Preserving Graph Reduction with Adaptive Structure for Persistent Homology

Abstract

Topological Data Analysis (TDA) provides artificial intelligence (AI) systems with mathematically rigorous geometric descriptors through Persistent Homology (PH), capturing essential shape characteristics in high-dimensional data. Yet, PH’s combinatorial complexity and sensitivity to outliers hinder its scalability and reliability, especially for Intrinsic PH (IPH) that relies on accurate geodesic distances. While stateof-the-art landmark-based subsampling methods, PH Landmarks, ameliorate computational costs and improve outlier robustness by selecting representative points based on local PH scores, it remain computationally intensive and at low sampling rates struggle to reconstruct the global topology. In this work, we introduce TOPOGRAPH, a simple yet powerful framework that preserves intrinsic topology. The resulting coarsened graph supports efficient IPH computations using Fermat distances. Experiments on both synthetic and realworld datasets show that TOPOGRAPH outperforms stateof-the-art sampling-based methods by achieving an order-ofmagnitude speedup and substantially improved topological fidelity in persistence diagrams, demonstrating its ability for robust and scalable topological data analysis.