Tensor train approximation of multivariate Gaussian density by scaling and squaring

Abstract

Tensor train decomposition is a promising tool for dealing with high dimensional arrays. Point mass filters utilise such arrays for representing probability density functions of the state. Proofs of concept of the application of the low rank decomposition have been provided in the literature. However, the application requires to design parameters, such as tensor train ranks. Since the parameters dictating the computational requirements are derived from the data according to more abstract hyper-parameters such as precision, an analysis of benchmark examples is needed for allocating resources. This paper studies the ranks in the case of Gaussian densities. The influence of correlation and the effect of rounding are discussed first. Efficiency of the density representation used by standard point mass filters is considered next. Aspects of series expansion of the Gaussian density evaluated over array are considered for the tensor train format. The growth of the ranks is illustrated on a four-dimensional example. An observation of the growth for a multi-dimensional case is made last. The lessons learned are valuable for designing efficient point mass filters. Namely, they show that at least the naive implementations of tensor decomposition methods do not break the curse of dimensionality.