Robust regularized least squares estimation in the presence of bounded data uncertainties

We study the problem of estimating an unknown deterministic signal that is observed through an unknown deterministic observation matrix and additive noise under the regularized residual error criterion. In this framework, we introduce a robust approach to this problem and consider the performance of an estimator relative to the performance of the least squares (LS) estimator tuned to the underlying unknown observation matrix and noise. This relative performance measure in fact turns out to be the regret of the estimator for not knowing the true parameters. Refraining from any statistical and structural assumptions both on the observation matrix and noise, we then minimize this regret with a robust LS estimation method, where we also demonstrate that this method can be cast as a semi-definite programming (SDP) problem. Numerical examples are also presented to illustrate the theoretical results.