Near-optimality of Linear Recovery in Gaussian Observation

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 ω = Ax + σξ of x corrupted by Gaussian noise ξ. 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 terms of its worstcase, over x ∈ X , expected ‖ · ‖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 X is given by a “separable” constraint like X = {x : ∑ i a 2 ix 2 i ≤ 1} or X = {x : maxi |aixi| ≤ 1}.