Mean-square estimation with high dimensional log-concave noise
Ankur A. Kulkarni · 2016
We consider the problem of mean-square estimation of the state of a discrete time dynamical system having additive non-Gaussian noise. We assume that the noise has the structure that at each time instant, it is a projection of a fixed high-dimensional noise vector with a log-concave density. We derive conditions which guarantee that, as the dimension of this noise vector grows large relative to the dimension of the state space, the observation space, and the time horizon, the optimal estimator of the problem with Gaussian noise becomes near-optimal for the problem with non-Gaussian noise. The results are derived by first showing an asymptotic Gaussian lower bound on the minimum error which holds even for nonlinear systems. This lower bound is shown to be asymptotically tight for linear systems. For nonlinear systems this bound is tight provided the noise has a strongly log-concave density with some additional structure. These results imply that estimates obtained by employing the Gaussian estimator on the non-Gaussian problem satisfy an approximate orthogonality principle. Moreover, the difference between the optimal estimate and the estimate derived from the Gaussian estimator vanishes strongly in L2. For linear systems, we get that the Kalman filter serves as a near-optimal estimator. A key ingredient in the proofs is a recent central limit theorem of Eldan and Klartag.