Error Analysis of Douglas-Rachford Algorithm for Linear Inverse Problems: Asymptotics of Proximity Operator for Squared Loss

Proximal splitting-based convex optimization is a promising approach to linear inverse problems because we can use some prior knowledge of the unknown variables explicitly. In this paper, we firstly analyze the asymptotic property of the proximity operator for the squared loss function, which appears in the update equations of some proximal splitting methods for linear inverse problems. The analysis shows that the output of the proximity operator can be characterized with a scalar random variable in the large system limit. Moreover, we investigate the asymptotic behavior of the Douglas-Rachford algorithm, which is one of the famous proximal splitting methods. From the asymptotic result, we can predict the evolution of the mean-square-error (MSE) in the algorithm for large-scale linear inverse problems. Simulation results demonstrate that the MSE performance of the Douglas-Rachford algorithm can be well predicted by the analytical result in compressed sensing with the $\ell_{1}$ optimization.