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- W2950535240 abstract "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 we seek an approximation to x0. When x0 has few nonzeros, useful approximations are obtained by l1-penalized l2 minimization, in which the reconstruction hxl solves 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 which n,Ntoinfty with n/N to delta and k/n to rho. Call the ratio MSE/sigma^2 the noise sensitivity. We develop formal expressions for the MSE of hxl, and evaluate its worst-case formal noise sensitivity over all types of k-sparse signals. The phase space 0 rhoMSE(delta). The phase boundary rho = rhoMSE(delta) is identical to the previously-known phase transition curve for equivalence of l1 - l0 minimization in the k-sparse noiseless case. Hence a single phase boundary describes the fundamental phase transitions both for the noiseless and noisy cases. Extensive computational experiments validate the predictions of this formalism, including the existence of game theoretical structures underlying it. Underlying our formalism is the AMP algorithm introduced earlier by the authors. Other papers by the authors detail expressions for the formal MSE of AMP and its close connection to l1-penalized reconstruction. Here we derive the minimax formal MSE of AMP and then read out results for l1-penalized reconstruction." @default.
- W2950535240 created "2019-06-27" @default.
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- W2950535240 date "2010-04-08" @default.
- W2950535240 modified "2023-09-27" @default.
- W2950535240 title "The Noise-Sensitivity Phase Transition in Compressed Sensing" @default.
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