Persistent magnitude homology on finite metric space

Abstract

Magnitude homology is an emerging approach that captures the intrinsic topological and geometric features of metric spaces. It offers a distinct theoretical lens for interpreting structural information within topological and geometric data analysis. This work introduces persistent magnitude homology, an extension of magnitude homology that captures multi-scale geometric and topological features of metric spaces. We construct the category of finite metric spaces with isometric embeddings and show that magnitude homology defines a functor to the category of abelian groups, naturally leading to the definition of persistent magnitude homology. We also introduce weighted persistent modules and weighted barcodes to offer both an algebraic and visual description of persistent magnitude homology. Additionally, we present an isometry theorem that relates interleaving distances and bottleneck distances, and establish stability results for persistent magnitude homology and magnitude profile. These results establish the stability of magnitude-based descriptors, bridging the gap between theory and practical application.