Precise Performance Analysis of the LASSO under Matrix Uncertainties
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In this paper, we consider the problem of recovering an unknown sparse signal _0 ∈R^n from noisy linear measurements = _0+ ∈R^m. A popular approach is to solve the ℓ_1-norm regularized least squares problem which is known as the LASSO. In many practical situations, the measurement matrix is not perfectely known and we only have a noisy version of it. We assume that the entries of the measurement matrix and of the noise vector are iid Gaussian with zero mean and variances 1/n and σ_^2. In this work, an imperfect measurement matrix is considered under which we precisely characterize the limiting behavior of the mean squared error and the probability of support recovery of the LASSO. The analysis is performed when the problem dimensions grow simultaneously to infinity at fixed rates. Numerical simulations validate the theoretical predictions derived in this paper.
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