$\ell^p$-Stability of Weighted Persistence Diagrams

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the $p$-edit distance for $p\in [1,\infty]$, and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the $p$-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the $p$-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts.