Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing

TL;DR

This paper establishes an explicit local linear convergence rate for ADMM applied to isotropic TV norm compressed sensing, supported by numerical experiments on large 3D and MRI data.

Contribution

It provides the first explicit local linear convergence rate analysis for ADMM in multi-dimensional isotropic TV compressed sensing problems.

Findings

Proven convergence rate is close to observed rates in numerical tests.

Numerical verification on large 3D and MRI data supports theoretical results.

Abstract

We prove an explicit local linear rate for ADMM solving the isotropic Total Variation (TV) norm compressed sensing problem in multiple dimensions, by analyzing the auxiliary variable in the equivalent Douglas-Rachford splitting on a dual problem. Numerical verification on large 3D problems and real MRI data will be shown. Though the proven rate is not sharp, it is close to the observed ones in numerical tests. The proven rate is not sharp, but it provides an explicit upper bound that appears close to the observed convergence rate in numerical experiments, although we do not claim this behavior holds in general.