Bayesian Covariance Modelling of Large Tensor-Variate Data Sets $\&$ Inverse Non-parametric Learning of the Unknown Model Parameter Vector

TL;DR

This paper introduces a Bayesian tensor-variate Gaussian Process framework for modeling high-dimensional tensor data and learning unknown model parameters, demonstrated through an astrophysical application.

Contribution

It develops a novel Bayesian approach using tensor-normal likelihoods and MCMC sampling for covariance modeling and parameter inference in tensor data.

Findings

Successful modeling of tensor covariance structures

Effective learning of unknown parameters with credible regions

Application to astrophysical data for Sun's location

Abstract

We present a method for modelling the covariance structure of tensor-variate data, with the ulterior aim of learning an unknown model parameter vector using such data. We express the high-dimensional observable as a function of this sought model parameter vector, and attempt to learn such a high-dimensional function given training data, by modelling it as a realisation from a tensor-variate Gaussian Process (GP). The likelihood of the unknowns given training data, is then tensor-normal. We choose vague priors on the unknown GP parameters (mean tensor and covariance matrices) and write the posterior probability density of these unknowns given the data. We perform posterior sampling using Random-Walk Metropolis-Hastings. Thereafter we learn the aforementioned unknown model parameter vector by performing posterior sampling in two different ways, given test and training data, using MCMC, to generate 95$\%$ HPD credible region on each unknown. We make an application of this method to the learning of the location of the Sun in the Milky Way disk.