Kriging in Tensor Train data format

TL;DR

This paper introduces a robust Tensor Train (TT) approximation method integrated with FFT-based Kriging, significantly reducing computational complexity for high-dimensional covariance matrix estimation.

Contribution

It combines TT approximation and FFT in Kriging, enabling efficient computation of covariance matrices in high dimensions with stable TT ranks.

Findings

Reduced Kriging computational complexity to O(d r^3 n)

TT ranks remain stable for increasing grid size and dimensions

Demonstrated advantages on synthetic and real datasets

Abstract

Combination of low-tensor rank techniques and the Fast Fourier transform (FFT) based methods had turned out to be prominent in accelerating various statistical operations such as Kriging, computing conditional covariance, geostatistical optimal design, and others. However, the approximation of a full tensor by its low-rank format can be computationally formidable. In this work, we incorporate the robust Tensor Train (TT) approximation of covariance matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is shown that here the computational complexity of Kriging is reduced to $\mathcal{O}(d r^3 n)$, where $n$ is the mode size of the estimation grid, $d$ is the number of variables (the dimension), and $r$ is the rank of the TT approximation of the covariance matrix. For many popular covariance functions the TT rank $r$ remains stable for increasing $n$ and $d$. The advantages of this approach against those using plain FFT are demonstrated in synthetic and real data examples.