图上的随机最优运输：框架和新的距离测度
Randomized Optimal Transport on a Graph: framework and new distance measures
摘要：
最近开发的路径袋（BoP）框架包括在图的所有可行路径上设置吉布斯-玻尔兹曼分布。这种概率分布更倾向于短路径而不是长路径，其中一个自由参数（温度$T$）控制着分布的熵水平。这种形式主义使新的距离或相异性的计算，最短路径和电阻之间的距离，这已被证明是执行聚类和分类任务。在这项工作中，袋的路径形式主义是通过添加两个独立的等式约束固定的开始和结束节点分布的路径（利润率）。当温度较低时，这种形式主义被证明是相当于一个放松的最佳运输问题的网络路径进行两个离散分布之间的节点流。通过考虑自由能最小化而不是传统的成本最小化来实现随机化。算法计算最佳自由能解决方案的两种类型的路径：击中（或吸收）的路径和非击中，定期，路径，并要求逆的$n \times n$矩阵与$n$是节点的数量。有趣的是，对于无向图上的规则路径，所得到的最优策略在确定性最优传输策略（$T \rightarrow 0^{+}$）和对应电路的解决方案（$T \rightarrow \infty$）之间插值。两个节点之间的距离措施和一组节点之间的不相似性，都集成节点上的权重，来自这个框架。
英文摘要：
The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature $T$) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins). When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths, and require the inversion of an $n \times n$ matrix with $n$ being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy ($T \rightarrow 0^{+}$) and the solution to the corresponding electrical circuit ($T \rightarrow \infty$). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
