Minimax subspace sampling in the presence of noise

We treat the problem of reconstructing a signal x that lies in a subspace W from its noisy samples. The samples are modelled as the inner products of x with a set of sampling vectors that span a subspace S, not necessarily equal to W. We consider two approaches to reconstructing x from the noisy samples: a least-squares (LS) method and a minimax mean-squared error (MSE) strategy. We show that if the elements of x are finite, then the minimax MSE approach results in a smaller MSE than the LS approach for all values of x. We then generalize the results to the problem of minimizing an inner-product MSE.