  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. An understanding of the behavior of the optimization
algorithms would be important for the tuning of the parameters and the
development of new algorithms. In this paper, we first 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
resultant conjecture, we can predict the evolution of the mean squared 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.
