We present GenusSink, a new class of approximate generalized Sinkhorn algorithms with shortest-path-distance costs for bounded genus (e.g. planar) graphs, providing near-linear time: (1) pre-processing, (2) iteration step, (3) final transport plan matrix querying and near-linear memory. Graphs handled by GenusSink include in particular planar graphs and bounded-genus meshes approximating 3D objects. GenusSink addresses total quadratic time complexity of its brute-force counterpart by leveraging separator-based decomposition of graphs, computational geometry techniques, and new results on fast matrix-vector multiplications with generalized distance matrices, using, in particular, Fourier analysis and low displacement rank theory. It is inspired by recent breakthroughs in graph theory on approximating bounded genus metrics with small treewidth metrics \citep{minor-free-paper}. The graph-centric approach enables us to target optimal transport problem with the corresponding distributions defined on the manifolds approximated by weighted graphs and with cost functions given by geodesic distances. We conduct rigorous theoretical analysis of GenusSink, provide practical implementations, leveraging newly introduced in this paper \textit{separation graph field integrators} (S-GFIs) data structures and present empirical verification. GenusSink provides orders of magnitude more accurate computations than other efficient Sinkhorn algorithms, while still guaranteeing significant computational improvements, as compared to the baseline. As a by-product of the developed methods, we show that GenusSink is \textbf{numerically equivalent} to the brute-force geodesic Sinkhorn algorithm on $n$-vertex graphs with treewidth $O(\log \log (n))$ (e.g. on trees).

翻译：我们提出GenusSink，一种针对有界属（例如平面）图、具有最短路径距离代价的新型近似广义Sinkhorn算法，实现了近线性复杂度：(1) 预处理，(2) 迭代步骤，(3) 最终运输计划矩阵查询，以及近线性内存。GenusSink处理的图特别包括平面图和近似三维物体的有界属网格。GenusSink通过利用基于分隔符的图分解、计算几何技术，以及在广义距离矩阵上快速矩阵-向量乘法的全新成果（特别地，使用傅里叶分析和低位移秩理论），解决了其暴力算法的大规模二次时间复杂性问题。该算法受近期图论中关于用小树宽度量近似有界属度量的突破性研究成果启发\citep{minor-free-paper}。这种以图为中心的方法使我们能够针对由加权图近似的流形上定义的对应分布、并由测地距离作为代价函数的最优传输问题。我们对GenusSink进行了严格的理论分析，提供了利用本文新引入的\textit{分隔图域积分器}（S-GFI）数据结构的实用实现，并给出了实证验证。与其他高效的Sinkhorn算法相比，GenusSink在提供数量级更精确计算的同时，仍保证相对于基线方法的显著计算改进。作为所开发方法的副产品，我们证明了GenusSink在树宽为$O(\log \log (n))$的$n$顶点图（例如树）上与暴力测地Sinkhorn算法\textbf{数值等价}。