Simple Error Bounds for Regularized Noisy Linear Inverse Problems

Consider estimating a structured signal $\mathbf{x}0$ from linear, underdetermined and noisy measurements $\mathbf{y}=\mathbf{A}\mathbf{x}0+\mathbf{z}$, via solving a variant of the lasso algorithm: $\hat{\mathbf{x}}=\arg\min\mathbf{x}{ |\mathbf{y}-\mathbf{A}\mathbf{x}|2+\lambda f(\mathbf{x})}$. Here, $f$ is a convex function aiming to promote the structure of $\mathbf{x}0$, say $\ell1$-norm to promote sparsity or nuclear norm to promote low-rankness. We assume that the entries of $\mathbf{A}$ are independent and normally distributed and make no assumptions on the noise vector $\mathbf{z}$, other than it being independent of $\mathbf{A}$. Under this generic setup, we derive a general, non-asymptotic and rather tight upper bound on the $\ell2$-norm of the estimation error $|\hat{\mathbf{x}}-\mathbf{x}0|2$. Our bound is geometric in nature and obeys a simple formula; the roles of $\lambda$, $f$ and $\mathbf{x}0$ are all captured by a single summary parameter $\delta(\lambda\partial((f(\mathbf{x}_0)))$, termed the Gaussian squared distance to the scaled subdifferential. We connect our result to the literature and verify its validity through simulations.