Near-optimality of linear recovery in Gaussian observation scheme under || ·||^sub 2^^sup 2^-loss

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 worst case, over x∈X, expected ||⋅||22-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:∑ia2ix2i≤1} or X={x:maxi|aixi|≤1}.