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.

翻译：我们改进并扩展了在~\cite{pegoraro2025persistence}中引入的持久球。持久球将上半平面上的一个可积测度$μ$（包括作为计数测度的持久图（PDs））映射到一个函数$S(μ)\in C(\mathbb{S}^2)$，并且该映射相对于1-Wasserstein部分传输距离$\mathrm{POT}_1$是稳定的。此外，据我们所知，持久球是拓扑机器学习中第一个显式表示，其逆映射在图像上的连续性在每个紧支撑目标处都得以建立。最近在部分传输空间中取得的有界基数双Lipschitz嵌入结果虽然强大，但并非由本文所考虑的这类显式汇总映射给出。我们的构造根植于凸几何：对于正测度，定义中的ReLU积分是提升Zonoid的支撑函数。基于~\cite{pegoraro2025persistence}，我们改进了定义以更好地匹配$\mathrm{POT}_1$的删除机制，通过带符号的对角线增广来编码部分传输。特别地，对于可积的$μ$，$S(0)$与$S(μ)$之间的一致范数仅取决于$μ$的持久性，无需任何特设的重新加权，这反映了以持久性为代价到对角线的最优传输。这产生了一个在测度级别上的无参数表示（直至数值离散化），同时能适应未来的扩展，其中$μ$是从PDs导出的平滑测度（例如，持久性强度函数~\citep{wu2024estimation}）。在涉及函数数据、时间序列、图、网格和点云的聚类、回归和分类任务中，更新后的持久球具有竞争力，并且通常优于持久图像、持久景观、持久样条和切片Wasserstein核基线。