Estimating structured signals in sparse noise: A precise noise sensitivity analysis

We consider the problem of estimating a structured signal xo from linear, underdetermined and noisy measurements y = Ax₀ + z, in the presence of sparse noise z. A natural approach to recovering x₀, that takes advantage of both the structure of xo and the sparsity of z is solving: x = arg min_x ||y - Ax||₁ subject to f(x) ≤ f(x₀) (constrained LAD estimator). Here, f is a convex function aiming to promote the structure of x₀, say ℓ₁-norm to promote sparsity or nuclear norm to promote low-rankness. We assume that the entries of A and the non-zero entries of z are i.i.d normal with variances 1 and σ², respectively. Our analysis precisely characterizes the asymptotic noise sensitivity ||x - x₀||²₂/σ² in the limit σ² → 0. We show analytically that the LAD method outperforms the more popular LASSO method when the noise is sparse. At the same time its performance is no more than π/2 times worse in the presence of non-sparse noise. Our simulation results verify the validity of our theoretical predictions.