On Multidimensional Optimal Estimators: Linearity Conditions

It is well-known that, when a multivariate Gaussian source is contaminated with Gaussian noise, a linear estimator minimizes the mean square estimation error, irrespective of the covariance matrices of both source and noise. This paper analyzes the conditions for linearity of optimal estimators for general source and noise distributions over vector spaces. Given a noise (or source) distribution, we derive conditions for existence and uniqueness of a matching source (or noise) distribution that renders the optimal estimator linear. Moreover, we establish a new characterization of the uniqueness of Gaussians: the multivariate Gaussian source-channel pair is the only pair for which the optimal estimator is linear at more than one signal-to-noise ratio.