AI 中文总结
该研究针对图上概率密度函数的最优传输问题，通过分析关联多面体提出最速下降算法，可高效计算1-Wasserstein距离，并将其应用于图的Ollivier-Ricci曲率计算。
AI 中文摘要
我们通过分析关联多面体，给出了一种用于确定定义在图\nt=(V,E)顶点上的两个给定概率密度函数之间传输距离的有效算法。该多面体的顶点对应图\nt中生成树的可行流，且该多面体的1-骨架是与图\nt上Glauber动力学相关的生成树状态图的投影。该传输问题的最优值，即1- Wasserstein距离，可通过沿该多面体顶点追踪传输成本来计算。我们证明传输成本的局部极小值也是全局极小值，这一结论催生了用于求解该传输问题的最速下降算法。若概率密度函数在δℤ（δ>0）上取离散值，则最优传输成本最多在(|V|-1)/δ步内达到。作为应用，我们给出了一种用于计算图的Ollivier-Ricci曲率的高效算法。
英文摘要
We give an effective algorithm for determining the transportation distance between two given probability density functions defined on the vertices of a graph $G=(V,E)$ by analyzing an associated polytope. The vertices of the polytope correspond to feasible flows on spanning trees in $G$, and the $1$-skeleton of the polytope is a projection of the spanning tree state graph associated with the Glauber dynamics on $G$. The optimal value of this transportation problem, known as the $1$-Wasserstein distance, can be computed by tracing the transportation cost along the vertices of this polytope. We show that a local minimum of the transportation cost is also a global minimum, and this leads to a steepest descent algorithm for solving the transportation problem. If the probability density functions take discrete values in $δ\mathbb{Z}$ for some $δ>0$, then the optimal transport cost can be reached in at most $\frac{|V|-1}δ$ steps. As an application, we give an efficient algorithm for computing the Ollivier--Ricci curvature of a graph.
Comments15 pages, 1 figure
