Aspects of density approximation by tensor trains

TL;DR

This paper investigates tensor train decompositions for density approximation in point-mass filters, addressing issues like negative values, correlation, and grid interpolation to improve high-dimensional Bayesian state estimation.

Contribution

It analyzes key challenges in tensor train density approximations, including negative values, correlation effects, and grid interpolation impacts, advancing the understanding of tensor methods in state estimation.

Findings

Negative density values can be mitigated with specific tensor approximations

Correlation influences tensor train ranks and approximation quality

Interpolating densities between grids affects accuracy and stability

Abstract

Point-mass filters solve Bayesian recursive relations by approximating probability density functions of a system state over grids of discrete points. The approach suffers from the curse of dimensionality. The exponential increase of the number of the grid points can be mitigated by application of low-rank approximations of multidimensional arrays. Tensor train decompositions represent individual values by the product of matrices. This paper focuses on selected issues that are substantial in state estimation. Namely, the contamination of the density approximations by negative values is discussed first. Functional decompositions of quadratic functions are compared with decompositions of discretised Gaussian densities next. In particular, the connection of correlation with tensor train ranks is explored. Last, the consequences of interpolating the density values from one grid to a new grid are analysed.