AI 中文总结
本文提出Probabilistic Ball Mapper，为Ball Mapper引入概率分布分配方案，构建软重叠矩阵与差异度量，实现Ball Mapper的概率值表示，可定量比较且保留几何可解释性与计算简便性。
AI 中文摘要
我们提出了Probabilistic Ball Mapper（概率Ball Mapper），这是Ball Mapper的一种形式化方法，其中每个数据点被分配仅以包含它的度量球为支撑的概率分布。这种分配既定义了从属于Ball Mapper覆盖的划分，也定义了从有限数据空间到该覆盖的马尔可夫核。我们研究两种分配方案：支撑集均匀规则，以及结合到地标距离同时保留基础覆盖的局部径向基规则。通过该核推送经验数据分布会生成顶点上的概率分布。从每个点态分布中条件独立地抽取两次会生成软重叠矩阵。该矩阵是对称、非负、半正定的，且以顶点分布为边缘分布，因此它提供了经典Ball Mapper重叠的质量归一化细化，而非另一种未归一化的边数。对于在公共覆盖上构建的图，顶点和重叠分布可直接比较；对于独立拟合的覆盖，我们构建了Wasserstein和融合Gromov-Wasserstein型差异，其考虑了顶点质量、公共 ambient 度量可用时的地标几何以及内在图关系。对于固定覆盖，我们推导了由分配规则的灵敏度、数据扰动的大小以及覆盖边界附近的数据质量控制的显式扰动界。当重新计算覆盖时，地标运动产生额外的变异源，对此我们提出基于传输的稳定性原理而非无条件定理。所得框架将Ball Mapper转化为概率值表示，适用于定量比较，同时保留其几何可解释性和计算简单性。
英文摘要
We introduce Probabilistic Ball Mapper, a formulation of Ball Mapper in which each data point is assigned a probability distribution supported only on the metric balls that contain it. This assignment defines both a partition subordinate to the Ball Mapper cover and a Markov kernel from the finite data space to the cover. We study two assignment schemes: a uniform-on-support rule and a localized radial-basis rule that incorporates distance to landmarks while preserving the underlying cover. Pushing the empirical data distribution through the kernel produces a probability distribution over vertices. Drawing twice, conditionally and independently, from each pointwise distribution produces a soft-overlap matrix. This matrix is symmetric, nonnegative, positive semidefinite, and has the vertex distribution as both marginals. It therefore provides a mass-normalized refinement of classical Ball Mapper overlap rather than another unnormalized edge count. For graphs constructed on a common cover, the vertex and overlap distributions can be compared directly. For independently fitted covers, we formulate Wasserstein and fused Gromov--Wasserstein-type discrepancies that account for vertex mass, landmark geometry when a common ambient metric is available, and intrinsic graph relations. For a fixed cover, we derive explicit perturbation bounds controlled by the sensitivity of the assignment rule, the magnitude of the data perturbation, and the data mass near cover boundaries. When the cover is recomputed, landmark motion creates an additional source of variation, for which we state a transport-based stability principle rather than an unconditional theorem. The resulting framework turns Ball Mapper into a probability-valued representation suitable for quantitative comparison while retaining its geometric interpretability and computational simplicity.
