Minimum MSE Estimation with Convex Constraints

We address the problem of minimum mean-squared error (MMSE) estimation under convex constraints. The familiar orthogonality principle, developed for linear constraints, is generalized to include convex restrictions. Using the extended principle, we study two types of convex constraints: constraints on the estimated vector (e.g. bounded norm), and constraints on the structure of the estimator (e.g. filter with bounded coefficients). It is shown that in both cases there exists a simple closed form expression for the constrained MMSE estimator. As an application of our approach, we develop Wiener type filters under certain restrictions, which allow for practical implementations.