Optimization of Generalized Mean-Square Error in Noisy Linear Estimation

A class of least squares problems that arises in linear Bayesian estimation is analyzed. The data vector y is given by the model y = P(Hθ + η) +w, where H is a known matrix, while θ, η, and w are uncorrelated random vectors. The goal is to obtain the best estimate for θ from the measured data. Applications of this estimation problem arise in multisensor data fusion problems and in wireless communication. The unknown matrix P is chosen to minimize the expected meansquared error E(‖θ − θ̂‖2) subject to a power constraint “trace (PP∗) ≤ P ,” where θ̂ is the best affine estimate of θ. Earlier work characterized an optimal P in the case where the noise term η vanished, while this paper analyzes the effect of η, assuming its covariance is a multiple of I. The singular value decomposition of an optimal P is expressed in the form VΣΠU∗ where V and U are unitary matrices related to the covariance of either θ or w, and singular vectors of H, Σ is diagonal, and Π is a permutation matrix. The analysis is carried out in two special cases: (i) H = I and (ii) covariance of θ is I. In case (i), Π does not depend on the power P . In case (ii), Π generally depends on P . The optimal Π is determined in the limit as the power tends to zero or infinity; a good approximation to an optimal Π is found for general P .