Asymptotic Analysis of LASSOs Solution Path with Implications for Approximate Message Passing
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This paper concerns the performance of the LASSO (also knows as basis pursuit denoising) for recovering sparse signals from undersampled, randomized, noisy measurements. We consider the recovery of the signal x_o ∈R^N from n random and noisy linear observations y= Ax_o + w, where A is the measurement matrix and w is the noise. The LASSO estimate is given by the solution to the optimization problem x_o with x̂_λ = _x 1/2y-Ax_2^2 + λx_1. Despite major progress in the theoretical analysis of the LASSO solution, little is known about its behavior as a function of the regularization parameter λ. In this paper we study two questions in the asymptotic setting (i.e., where N →∞, n →∞ while the ratio n/N converges to a fixed number in (0,1)): (i) How does the size of the active set x̂_λ_0/N behave as a function of λ, and (ii) How does the mean square error x̂_λ - x_o_2^2/N behave as a function of λ? We then employ these results in a new, reliable algorithm for solving LASSO based on approximate message passing (AMP).
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