Tensor rank reduction via coordinate flows

TL;DR

This paper introduces a novel tensor rank reduction method using coordinate transformations, specifically linear ones, to improve the efficiency of high-dimensional tensor approximations for PDEs.

Contribution

It proposes a new tensor rank reduction technique based on linear coordinate transformations and develops an algorithm using Riemannian gradient descent for this purpose.

Findings

Effective tensor rank reduction demonstrated on PDE problems

Algorithm achieves near-optimal linear coordinate transformations

Potential for extension to nonlinear transformations

Abstract

Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has smaller tensor rank. We restrict our analysis to linear coordinate transformations, which gives rise to a new class of functions that we refer to as tensor ridge functions. Leveraging Riemannian gradient descent on matrix manifolds we develop an algorithm that determines a quasi-optimal linear coordinate transformation for tensor rank reduction.The results we present for rank reduction via linear coordinate transformations open the possibility for generalizations to larger classes of nonlinear transformations. Numerical applications are presented and discussed for linear and nonlinear PDEs.