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Proximal splitting-based convex optimization is a promising approach tolinear inverse problems because we can use some prior knowledge of the unknownvariables explicitly. An understanding of the behavior of the optimizationalgorithms would be important for the tuning of the parameters and thedevelopment of new algorithms. In this paper, we first analyze the asymptoticproperty of the proximity operator for the squared loss function, which appearsin the update equations of some proximal splitting methods for linear inverseproblems. The analysis shows that the output of the proximity operator can becharacterized with a scalar random variable in the large system limit.Moreover, we investigate the asymptotic behavior of the Douglas-Rachfordalgorithm, which is one of the famous proximal splitting methods. From theresultant conjecture, we can predict the evolution of the mean squared error(MSE) in the algorithm for large-scale linear inverse problems. Simulationresults demonstrate that the MSE performance of the Douglas-Rachford algorithmcan be well predicted by the analytical result in compressed sensing with the$\ell_{1}$ optimization.

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