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.

翻译：近年来，基于张量网络和低秩技术的高效数值算法在近似高维函数及高维偏微分方程解方面引起了广泛关注。本文提出了一种基于坐标变换的新型张量秩缩减方法，可显著提升高维张量近似算法的效率。核心思想简洁明了：给定一个多元函数，确定一个坐标变换，使得新坐标系下的函数具有更小的张量秩。本研究将分析限定于线性坐标变换，由此衍生出一类新型函数——我们称之为张量脊函数。通过利用矩阵流形上的黎曼梯度下降方法，我们开发了一种算法，可确定用于张量秩缩减的准最优线性坐标变换。本文展示的线性坐标变换秩缩减结果，为推广至更广义的非线性变换类别奠定了基础。最后，我们针对线性和非线性偏微分方程给出了数值应用实例并进行了讨论。