Paper Info
Title
Mean-Square Estimation With High Dimensional Log-Concave Noise
Abstract
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 L-2. 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.
Year
Venue
Field
2016
2016 IEEE 55TH CONFERENCE ON DECISION AND CONTROL (CDC)
Mathematical optimization,Gaussian random field,Linear-quadratic-Gaussian control,Gaussian noise,Gaussian function,Additive white Gaussian noise,Mathematics,Estimator,Gradient noise,Orthogonality principle
DocType
ISSN
Citations 
Conference
0743-1546
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Ankur A. Kulkarni110620.95