Persistence Spheres: Bi-continuous Representations of Persistence Diagrams
Matteo Pegoraro
TL;DR
Persistence spheres provide a bi-continuous functional embedding of persistence diagrams into functions on $\mathbb{S}^2$, achieving Lipschitz stability with respect to the $1$-Wasserstein distance and an interpretable, explicit inverse on their image. Built via the lift zonoid of weighted PDs, PSs yield a linear, separable representation with a closed-form expression $\varphi_{\mu_D}^{\omega}(v)=\sum_{p\in D}\omega(p)c_p\mathrm{ReLU}(\langle v,(1,p)\rangle)$, ensuring both stability and geometric fidelity to the Wasserstein geometry. The authors establish continuity theorems linking convergence of lift zonoids to convergence in $W_1$, and show PSs deliver competitive or superior performance across regression and classification tasks on diverse data types, while allowing parallel, scalable computation. The framework opens avenues for richer statistical tools in topological data analysis, such as confidence sets and hypothesis tests via its functional representation, and suggests extensions to signed measures or alternative weightings to tailor expressivity. Overall, persistence spheres advance topological machine learning by offering a principled, scalable, and faithful embedding of persistence information into linear-like spaces.
Abstract
We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.