Asymptotic Analysis of ADMM for Compressed Sensing
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In this paper, we analyze the asymptotic behavior of alternating direction method of multipliers (ADMM) for compressed sensing, where we reconstruct an unknown structured signal from its underdetermined linear measurements. The analytical tool used in this paper is recently developed convex Gaussian min-max theorem (CGMT), which can be applied to various convex optimization problems to obtain its asymptotic error performance. In our analysis of ADMM, we analyze the convex subproblem in the update of ADMM and characterize the asymptotic distribution of the tentative estimate obtained at each iteration. The result shows that the update equations in ADMM can be decoupled into a scalar-valued stochastic process in the asymptotic regime with the large system limit. From the asymptotic result, we can predict the evolution of the error (e.g. mean-square-error (MSE) and symbol error rate (SER)) in ADMM for large-scale compressed sensing problems. Simulation results show that the empirical performance of ADMM and its theoretical prediction are close to each other in sparse vector reconstruction and binary vector reconstruction.
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