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 = Ax
₀
+ z, in the presence of sparse noise z. A natural approach to recovering x
₀
, 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(x
₀
) (constrained LAD estimator). Here, f is a convex function aiming to promote the structure of x
₀
, say ℓ
₁
-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 σ
²
, respectively. Our analysis precisely characterizes the asymptotic noise sensitivity ||x - x
₀
||
²
₂
/σ
²
in the limit σ
²
→ 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.