Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

Abstract

We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $μ$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function $S(μ)\in C(\mathbb{S}^2)$, and the map is stable with respect to 1-Wasserstein partial transport distance $\mathrm{POT}_1$. Moreover, to the best of our knowledge, persistence spheres are the first explicit representation used in topological machine learning for which continuity of the inverse on the image is established at every compactly supported target. Recent bounded-cardinality bi-Lipschitz embedding results in partial transport spaces, despite being powerful, are not given by the kind of explicit summary map considered here. Our construction is rooted in convex geometry: for positive measures, the defining ReLU integral is the support function of the lift zonoid. Building on~\cite{pegoraro2025persistence}, we refine the definition to better match the $\mathrm{POT}_1$ deletion mechanism, encoding partial transport via a signed diagonal augmentation. In particular, for integrable $μ$, the uniform norm between $S(0)$ and $S(μ)$ depends only on the persistence of $μ$, without any need of ad-hoc re-weightings, reflecting optimal transport to the diagonal at persistence cost. This yields a parameter-free representation at the level of measures (up to numerical discretization), while accommodating future extensions where $μ$ is a smoothed measure derived from PDs (e.g., persistence intensity functions~\citep{wu2024estimation}). Across clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds, the updated persistence spheres are competitive and often improve upon persistence images, persistence landscapes, persistence splines, and sliced Wasserstein kernel baselines.

Figures (6)

• Figure 1: Example of the lift zonoid construction for a discrete measure $\mu=\sum_i c_i\delta_{p_i}$. Each atom contributes the segment $c_i[0,(1,p_i)]\subset\mathbb{R}^3$, and $Z_\mu$ is obtained as their Minkowski sum.
• Figure 2: One-point diagrams drifting along $(1,1)$. Left: attenuation of a pure diagonal offset between $D_k$ and $D_k'$. Right: behavior of the deletion-to-diagonal cost for $D_k$. The updated definition matches the $\mathop{\mathrm{POT}}\nolimits_1$ geometry by construction, whereas in pegoraro2025persistence both effects are mediated by the decay of the reweighting $\omega_K$ along diagonal lines.
• Figure 3: Two elementary mechanisms by which standard topological summaries deform the $\mathop{\mathrm{POT}}\nolimits_1$ geometry. Persistence landscapes amplify variability at high persistence, while Gaussian persistence images induce a kernel-dependent saturation of pairwise distances. In the latter case, for fixed bandwidth $\sigma$, the discrepancy with $\mathop{\mathrm{POT}}\nolimits_1$ becomes more pronounced at high persistence, since the persistence-image distance saturates at a fixed kernel scale whereas the $\mathop{\mathrm{POT}}\nolimits_1$ flattening occurs only at the much larger deletion scale determined by the sum of the persistences.
• Figure 4: FDA simulations. Left: realizations from the baseline generative model used in scenarios (i)--(iii), where the discriminative signal is more likely to be carried by large oscillations while noise induces abundant low-persistence clutter. Right: realizations from scenario (iv), where the largest oscillations are shared across the two classes, so that an excessive bias toward high-persistence features becomes detrimental.
• Figure 5: PP simulation. Top row: one simulated point pattern for each point process. Bottom row: corresponding persistence diagrams in degrees $H_0$ (blue) and $H_1$ (red). Ordering: Poisson, Thomas, Matérn, jittered lattice.

Theorems & Definitions (50)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Theorem 1: Convergence for $\mathop{\mathrm{POT}}\nolimits_1$
• Definition 8