Near-optimality of linear recovery in Gaussian observation scheme under $\Vert \cdot \Vert_{2}^{2}$-loss

Abstract

We consider the problem of recovering linear image $Bx$ of a signal $x$ known to belong to a given convex compact set $\mathcal{X}$ from indirect observation $\omega=Ax+\sigma\xi$ of $x$ corrupted by Gaussian noise $\xi$. It is shown that under some assumptions on $\mathcal{X}$ (satisfied, e.g., when $\mathcal{X}$ is the intersection of $K$ concentric ellipsoids/elliptic cylinders), an easy-to-compute linear estimate is near-optimal in terms of its worst case, over $x\in\mathcal{X}$, expected $\Vert \cdot \Vert_{2}^{2}$-loss. The main novelty here is that the result imposes no restrictions on $A$ and $B$. To the best of our knowledge, preceding results on optimality of linear estimates dealt either with one-dimensional $Bx$ (estimation of linear forms) or with the “diagonal case” where $A$, $B$ are diagonal and $\mathcal{X}$ is given by a “separable” constraint like $\mathcal{X}=\{x:\sum_{i}a_{i}^{2}x_{i}^{2}\leq1\}$ or $\mathcal{X}=\{x:\max_{i}|a_{i}x_{i}|\leq1\}$.