  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.
