Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size

TL;DR

This paper introduces a primal-dual optimization method that efficiently solves large-scale L1-type problems with convergence rates unaffected by grid size, enabling rapid processing of high-resolution images and transport problems.

Contribution

The paper presents a novel primal-dual algorithm with grid-size independent convergence, applicable to large-scale image denoising and optimal transport problems.

Findings

Converges independently of grid size.

Solves 4096x4096 problems in minutes.

Applicable to image denoising and optimal transport.

Abstract

We present a primal-dual method to solve L1-type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 by 4096 in a few minutes without parallelization.