Matteo Pegoraro: Persistence Spheres: Bi-Continuous Representations of Persistence Diagrams

Abstract.

Persistence spheres are a new functional representation of persistence diagrams. We show that they induce a bi-continuous mapping: PSs are Lipschitz with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This yields, in a theoretically optimal sense, both stability and geometric fidelity, since bi-Lipschitz embeddings are provably impossible in this setting. We derive explicit formulas for PSs and show that they can be computed efficiently. Empirically, we evaluate them on clustering, regression, and classification tasks involving diverse data types, and we provide practical guidance for tuning their parameters.