🤖 AI Summary
This work addresses the lack of stability for large graded persistent barcodes in finite pseudo-metric spaces. Methodologically, it introduces bifiltered persistent double homology theory: starting from the Vietoris–Rips filtration, it constructs the associated projective angular complex and defines both ordinary and double homology—yielding bifiltered persistent homology modules and barcodes. The paper establishes, for the first time, an $L^infty$-stability theorem for this bifiltered structure, rigorously proving the robustness of both modules and barcodes under perturbations of the input metric. By extending persistent homology from the conventional single-parameter (i.e., singly graded) setting to a two-parameter (bifiltered, doubly graded) framework, this work provides a novel algebraic-topological foundation and theoretical guarantee for high-dimensional topological data analysis—particularly for multi-scale and multiparameter feature extraction.
📝 Abstract
We define bigraded persistent homology modules and bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris-Rips filtration of X. We prove a stability theorem for the bigraded persistent double homology modules and barcodes.