Nearest-Neighbor Interaction Systems in the Tensor-Train Format

TL;DR

This paper introduces a tensor-train decomposition method for nearest-neighbor interaction systems, significantly reducing memory and computational costs, enabling efficient analysis of high-dimensional problems like the Ising model and coupled oscillators.

Contribution

It presents a systematic tensor-train approach applicable to various problems, with ranks that can be independent of network size, facilitating high-dimensional system analysis.

Findings

Memory consumption is significantly reduced.

Computational costs are lowered substantially.

Tensor ranks can be independent of system size.

Abstract

Low-rank tensor approximation approaches have become an important tool in the scientific computing community. The aim is to enable the simulation and analysis of high-dimensional problems which cannot be solved using conventional methods anymore due to the so-called curse of dimensionality. This requires techniques to handle linear operators defined on extremely large state spaces and to solve the resulting systems of linear equations or eigenvalue problems. In this paper, we present a systematic tensor-train decomposition for nearest-neighbor interaction systems which is applicable to a host of different problems. With the aid of this decomposition, it is possible to reduce the memory consumption as well as the computational costs significantly. Furthermore, it can be shown that in some cases the rank of the tensor decomposition does not depend on the network size. The format is thus feasible even for high-dimensional systems. We will illustrate the results with several guiding examples such as the Ising model, a system of coupled oscillators, and a CO oxidation model.