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，一种针对有界亏格（例如平面）图、具有最短路径距离代价的近似广义Sinkhorn新算法族，实现了近线性时间的：(1)预处理、(2)迭代步骤、(3)最终传输计划矩阵查询，以及近线性内存存储。GenusSink特别适用于平面图和逼近3D物体的有界亏格网格。该算法通过利用基于分隔符的图分解、计算几何技术，以及基于傅里叶分析和低位移秩理论的新型广义距离矩阵快速矩阵向量乘法结果，解决了暴力对应算法的总二次时间复杂度问题。其灵感源于近期图论中利用小树宽度量逼近有界亏格度量的突破性成果\citep{minor-free-paper}。这种以图为中心的方法使我们能够处理定义在由加权图逼近的流形上的对应分布、以测地距离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{分离图场积分器}（S-GFIs）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在计算精度上提升了数个数量级，同时相比基线方法仍保证了显著的计算性能改进。作为所开发方法的副产品，我们证明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}。