Near-Optimality of Linear Recovery in Gaussian Observation Scheme under $\|\cdot\|_2^2$-Loss

Abstract

We consider the problem of recovering linear image $Bx$ of a signal $x$ known to belong to a given convex compact set $X$ from indirect observation $\omega=Ax+\sigma\xi$ of $x$ corrupted by Gaussian noise $\xi$. It is shown that under some assumptions on $X$ (satisfied, e.g., when $X$ is the intersection of $K$ concentric ellipsoids/elliptic cylinders), an easy-to-compute linear estimate is near-optimal, in certain precise sense, in terms of its worst-case, over $x\in X$, expected $\|\cdot\|_2^2$-error. The main novelty here is that our results impose no restrictions on $A$ and $B$, to the best of our knowledge, preceding results on optimality of linear estimates dealt either with the case of direct observations $A=I$ and $B=I$, or with the "diagonal case" where $A$, $B$ are diagonal and $X$ is given by a "separable" constraint like $X=\{x:\sum_ia_i^2x_i^2\leq 1\}$ or $X=\{x:\max_i|a_ix_i|\leq1\}$, or with estimating a linear form (i.e., the case one-dimensional $Bx$).