Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing

TL;DR

The paper studies isotropic TV norm compressed sensing with undersampled Fourier measurements and analyzes ADMM via its dual Douglas–Rachford formulation. It shows ADMM is equivalent to G-prox PDHG and derives an explicit local linear convergence rate for multidimensional TVCS, expressed in terms of the smallest nonzero principal angle $\theta_1$ between $\mathcal{K}\operatorname{Kernel}(A)$ and $\operatorname{Kernel}(\widetilde{B})$ and a dimension-dependent factor involving the dual fixed point $v_*$ and step size $\tau$. Under a random sampling assumption $|\Omega| \ge C_s^{-1} |\mathcal{S}|\log(N)$, the minimizer is unique with high probability and the fixed point is interior, enabling the linear rate to hold locally. Numerical experiments on 2D and 3D TVCS problems, including MRI data, confirm the predicted linear regime and illustrate robustness to step-size choices and to single- vs. double-precision arithmetic, highlighting practical implications for large-scale imaging applications.

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.