Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing
This provides theoretical guarantees for a specific optimization method in compressed sensing, but it is incremental as it builds on existing ADMM and TV norm frameworks.
The paper tackles the problem of proving convergence rates for ADMM in solving isotropic Total Variation norm compressed sensing, and shows an explicit local linear rate that is close to observed rates in numerical tests on 3D and MRI data.
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.