Asymptotic Analysis of MAP Estimation via the Replica Method and Compressed Sensing

The replica method is a non-rigorous but widely-used technique from statistical physics used in the asymptotic analysis of many large random nonlinear problems. This paper applies the replica method to non-Gaussian MAP estimation. It is shown that with large random linear measurements and Gaussian noise, the asymptotic behavior of the MAP estimate of an n-dimensional vector ``decouples as n scalar MAP estimators. The result is a counterpart to Guo and Verdus replica analysis on MMSE estimation. The replica MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding and zero-norm estimation. In the case of lasso estimation, the scalar estimator reduces to a soft-thresholding operator and for zero-norm estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally tractable method for exactly computing various performance metrics including MSE and sparsity recovery.