Near-Linear Time Generalized Sinkhorn Algorithms for Bounded Genus Graphs

TL;DR

GenusSink is a near-linear time approximate algorithm for optimal transport on bounded genus graphs, leveraging graph decomposition, geometry, and new data structures for efficient computation.

Contribution

It introduces GenusSink, a novel near-linear time algorithm for optimal transport on bounded genus graphs, combining separator techniques and advanced matrix multiplication methods.

Findings

GenusSink achieves significantly more accurate results than existing Sinkhorn algorithms.

The algorithm operates in near-linear time with practical implementation and empirical validation.

GenusSink is numerically equivalent to brute-force geodesic Sinkhorn on certain graph classes.

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).