On the convergence and sampling of randomized primal-dual algorithms and their application to parallel MRI reconstruction

TL;DR

This paper provides a theoretical analysis of the convergence of the stochastic primal-dual hybrid gradient algorithm and explores its application to parallel MRI reconstruction with various sampling strategies.

Contribution

It proves the almost sure convergence of SPDHG for convex, non-strongly convex, and non-smooth functions under arbitrary random sampling, and evaluates its performance in MRI reconstruction.

Findings

Sampling strategies significantly influence convergence speed.

Optimal sampling methods can be identified for better performance.

SPDHG effectively reconstructs MRI images with appropriate sampling.

Abstract

Stochastic Primal-Dual Hybrid Gradient (SPDHG) is an algorithm proposed by Chambolle et al. (2018) to efficiently solve a wide class of nonsmooth large-scale optimization problems. In this paper we contribute to its theoretical foundations and prove its almost sure convergence for convex but neither necessarily strongly convex nor smooth functionals, as well as for any random sampling. In addition, we study SPDHG for parallel Magnetic Resonance Imaging reconstruction, where data from different coils are randomly selected at each iteration. We apply SPDHG using a wide range of random sampling methods and compare its performance across a range of settings, including mini-batch size and step size parameters. We show that the sampling can significantly affect the convergence speed of SPDHG and for many cases an optimal sampling can be identified.