Inexact Bregman Sparse Newton Method for Efficient Optimal Transport

TL;DR

The paper introduces the Inexact Bregman Sparse Newton method, a novel approach for efficiently solving large-scale exact optimal transport problems with high accuracy and reduced computational costs.

Contribution

It proposes a new inexact Newton-based algorithm with sparsification techniques for fast, precise solutions to optimal transport, overcoming limitations of existing methods.

Findings

Outperforms state-of-the-art methods in speed and accuracy

Maintains convergence guarantees despite inexact subproblem solutions

Reduces memory and computational costs significantly

Abstract

Computing exact Optimal Transport (OT) distances for large-scale datasets is computationally prohibitive. While entropy-regularized alternatives offer speed, they sacrifice precision and frequently suffer from numerical instability in high-accuracy regimes. To address these limitations, we propose the Inexact Bregman Sparse Newton (IBSN) method, which efficiently solves the exact OT problems. Our approach utilizes a Bregman proximal point framework through a sequence of semi-dual subproblems. By solving these subproblems inexactly, we significantly reduce per-iteration complexity while maintaining a theoretical guarantee of convergence to the true optimal plan. To further accelerate the algorithm, we develop a sparse Newton-type solver for the subproblem and employ a Hessian sparsification strategy that drastically lowers memory and time costs without sacrificing accuracy. We provide rigorous theoretical guarantees for the global convergence of the algorithm. Extensive experiments demonstrate that IBSN consistently outperforms state-of-the-art methods in both computational speed and solution precision.