PERSISTENCE SPHERES: BI-CONTINUOUS REPRESENTATIONS OF PERSISTENCE DIAGRAMS.

Persistence spheres are a new functional representation of persistence diagrams. In contrast to existing embeddings such as persistence images, landscapes, or kernel-based methods, persistence spheres define a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This provides, in a theoretically optimal sense, both stability and geometric fidelity, so that persistence spheres are the representation that most closely preserves the Wasserstein geometry of PDs in a linear space. We derive explicit formulas for persistence spheres, show that they can be computed efficiently, and note that they parallelize with minimal overhead. Empirically, we evaluate them on clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently achieve state-of-the-art or competitive performance when compared with persistence images, persistence landscapes, persistence splines, and the sliced Wasserstein kernel. Additional simulations in the appendices further support the method and provide practical guidance for tuning its parameters.