Near-Linear Time Generalized Sinkhorn Algorithms for Bounded Genus Graphs

2026-05-10

AI Summary

This work addresses the quadratic computational bottleneck of the classical Sinkhorn algorithm for optimal transport based on geodesic distances over bounded-genus graphs, such as planar graphs. The authors propose GenusSink, the first algorithm that integrates the structural properties of bounded-genus graphs into the Sinkhorn framework. By introducing a Separator-based Graph Field Integrator (S-GFI) data structure and combining graph separator decomposition, low-shift-rank approximation, Fourier analysis, and computational geometry techniques, GenusSink achieves near-linear time and memory complexity across preprocessing, iterative updates, and query phases. On low-treewidth graphs such as trees, the method is numerically equivalent to brute-force computation, while significantly outperforming existing efficient Sinkhorn variants in both speed and accuracy—yielding improvements of several orders of magnitude in precision.

Abstract

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).

Problem

• optimal transport
• bounded genus graphs
• Sinkhorn algorithm
• geodesic distance
• computational complexity

Innovation

• GenusSink
• bounded genus graphs
• near-linear time
• separator decomposition
• optimal transport