Functorial Topological Data Compression via Stratified Persistent Sheaves and Enriched Interleavings

Abstract

We develop a comprehensive sheaf-theoretic framework for topological data analysis in which the primary invariant is a stratified persistent sheaf on a filtered space, functorial with respect to both refinement of stratification and restriction to sublevel sets. Our approach remedies fundamental limitations of classical persistent homology by localizing information across strata and enabling the detection of features tied to specific regions of a data manifold. Working concretely with constructible sheaves of finite-dimensional vector spaces on Whitney-stratified spaces, we define an enriched interleaving distance that incorporates stratum-wise equivalence criteria, proving that it yields an extended metric on the category of stratified persistent sheaves. The central theoretical contribution is a stability theorem: perturbations of the filtration function induce controlled perturbations in the enriched interleaving distance, with explicit bounds involving the sup-norm of the perturbation.