Simple Error Bounds for Regularized Noisy Linear Inverse Problems

Consider estimating a structured signal x_0 from linear, underdetermined and noisy measurements y = Ax_0+z, via solving a variant of the lasso algorithm: x̂ = arg min_x{∥y-Ax∥+2 + λf(x)}. 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 are independent and normally distributed and make no assumptions on the noise vector z, other than it being independent of A. Under this generic setup, we derive a general, non-asymptotic and rather tight upper bound on the ℓ_2-norm of the estimation error ∥x̂ - x0∥2. Our bound is geometric in nature and obeys a simple formula; the roles of λ, f and x_0 are all captured by a single summary parameter δ(λ∂f(x_0)), termed the Gaussian squared distance to the scaled subdifferential. We connect our result to the literature and verify its validity through simulations.