Annealed Sinkhorn with Momentum: Certified Unregularized Optimal Transport in Linear Memory

Abstract

We characterize Bregman Douglas-Rachford splitting (BDRS) for unregularized discrete optimal transport and develop an anytime primal-dual certificate in linear memory. We first establish that BDRS coincides with warm-started Inexact Proximal point method for exact Optimal Transport (IPOT) using a single inner Sinkhorn iteration. By eliminating the primal transport plan from the updates, we derive an equivalent dual formulation that reveals BDRS as annealed Sinkhorn under an implicit inverse-linear temperature schedule, with an additional log-scaling momentum term and a cooler kernel. While this explains the role of the temperature parameter in BDRS as an initial temperature, it also reduces the solver's memory requirement from quadratic to linear. Utilizing this annealing perspective, we introduce overrelaxed BDRS, which combines annealing and overrelaxed scaling within a single recursion. We derive a primal-dual certificate for both methods that can be evaluated in linear memory without transport plan construction, thus providing a computable stopping rule. On the DOTmark benchmark, combining momentum with the cooler kernel produces substantially smaller optimality gaps than annealed Sinkhorn under the same schedule. For pixel-level color transfer between $1024\times1024$ images, BDRS attains a lower repaired transport cost than MDOT-TNT with a 9$\times$ speed up, reaching a relative duality gap of $1.59\%$ in 24 minutes. We further demonstrate a color transfer with $4238\times2365$ images, yielding 10 million pixels per image and approximately one hundred trillion implicit transport entries, reaching a best relative duality gap of $2.41\%$ and $2.80\%$ within 35 hours in each direction on a single NVIDIA L40S GPU.