Intrinsic Multiresolution Persistent Homology: Diffusion-Induced Resolutions and Entropy-Summary Representations

In topological data analysis, persistent homology (PH) characterizes topological features across filtration scales. Conventional PH, however, analyzes a single geometric representation across these scales and therefore does not organize features by intrinsic geometric resolution. This creates two coupled gaps: resolutions are fixed in advance rather than data-induced, and separate summaries do not jointly retain resolution and filtration-domain information. We introduce (Diffusion-Induced Geometry for Intrinsic-Resolution Topology), which constructs ordered diffusion-induced geometries and selects each sample's retained range from its diffusion evolution. Its multiresolution entropy summary function (\MESF) integrates the resulting persistence diagrams while retaining diffusion resolution, filtration-domain information, and persistence contribution. We establish conditional stability under discrete diffusion-operator perturbations. Experiments demonstrate progressive topological coarse-graining on synthetic point clouds; on three CATH benchmarks at the Class, Architecture, and Topology levels, + attains the highest mean accuracy among the evaluated persistence-based representations on all benchmarks.