Jensen Bregman LogDet Divergence Optimal Filtering in the Manifold of Positive Definite Matrices

摘要

In this paper, we consider the problem of optimal estimation of a time-varying positive definite matrix from a collection of noisy measurements. We assume that this positive definite matrix evolves according to an unknown GARCH (generalized auto-regressive conditional het-eroskedasticity) model whose parameters must be estimated from experimental data. The main difficulty here, compared against traditional parameter estimation methods, is that the estimation algorithm should take into account the fact that the matrix evolves on the PD manifold. As we show in the paper, measuring the estimation error using the Jensen Bregman LogDet divergence leads to computationally tractable (and in many cases convex) problems that can be efficiently solved using first order methods. Further, since it is known that this metric provides a good surrogate of the Riemannian manifold metric, the resulting algorithm respects the non-Euclidean geometry of the manifold. In the second part of the paper we show how to exploit this model in a maximum likelihood setup to obtain optimal estimates of the unknown matrix. In this case, the use of the JBLD metric allows for obtaining an alternative representation of Gaussian conjugate priors that results in closed form solutions for the maximum likelihood estimate. In turn, this leads to computationally efficient algorithms that take into account the non-Euclidean geometry. These results are illustrated with several examples using both synthetic and real data.