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Consider the noisy underdetermined system of linear equations: y=Ax0 + z0,with n x N measurement matrix A, n < N, and Gaussian white noise z0 ~N(0,\sigma^2 I). Both y and A are known, both x0 and z0 are unknown, and weseek an approximation to x0. When x0 has few nonzeros, useful approximationsare obtained by l1-penalized l2 minimization, in which the reconstruction \hxlsolves min || y - Ax||^2/2 + \lambda ||x||_1.  Evaluate performance by mean-squared error (MSE = E ||\hxl - x0||_2^2/N).Consider matrices A with iid Gaussian entries and a large-system limit in whichn,N\to\infty with n/N \to \delta and k/n \to \rho. Call the ratio MSE/\sigma^2the noise sensitivity. We develop formal expressions for the MSE of \hxl, andevaluate its worst-case formal noise sensitivity over all types of k-sparsesignals. The phase space 0 < \delta, \rho < 1 is partitioned by curve \rho =\rhoMSE(\delta) into two regions. Formal noise sensitivity is boundedthroughout the region \rho < \rhoMSE(\delta) and is unbounded throughout theregion \rho > \rhoMSE(\delta). The phase boundary \rho = \rhoMSE(\delta) isidentical to the previously-known phase transition curve for equivalence of l1- l0 minimization in the k-sparse noiseless case. Hence a single phase boundarydescribes the fundamental phase transitions both for the noiseless and noisycases. Extensive computational experiments validate the predictions of thisformalism, including the existence of game theoretical structures underlyingit. Underlying our formalism is the AMP algorithm introduced earlier by theauthors. Other papers by the authors detail expressions for the formal MSE ofAMP and its close connection to l1-penalized reconstruction. Here we derive theminimax formal MSE of AMP and then read out results for l1-penalizedreconstruction.

The Noise-Sensitivity Phase Transition in Compressed Sensing