Convergence Properties of a Randomized Primal-Dual Algorithm with Applications to Parallel MRI

TL;DR

This paper proves the almost sure convergence of the SPDHG algorithm for convex problems and demonstrates its superior speed in parallel MRI reconstruction compared to deterministic methods.

Contribution

It establishes the convergence properties of SPDHG for convex functions and applies it successfully to parallel MRI, showing improved performance.

Findings

SPDHG converges faster than deterministic algorithms in MRI reconstruction.

Proved almost sure convergence of SPDHG for convex, not necessarily strongly convex functions.

Numerical results confirm the efficiency of SPDHG in practical applications.

Abstract

The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.