论文标题
通过坐标流量降低张量排名
Tensor rank reduction via coordinate flows
论文作者
论文摘要
最近，人们对基于张量网络和低级技术的有效数值算法的兴趣越来越大，以近似高维函数和对高度PDES的解决方案。在本文中，我们提出了一种基于坐标转换的新张量降低方法，该方法可以大大提高高维张量近似算法的效率。这个想法很简单：给定一个多元函数，确定坐标转换，以使新坐标系中的函数具有较小的张量排名。我们将分析限制为线性坐标转换，这导致了新的功能，我们称为张量脊函数。利用riemannian梯度下降在矩阵歧管上，我们开发了一种算法，该算法确定了张量级降低的准最佳线性坐标转换。我们通过线性坐标转换提供了秩降低的结果，为对非线性转换的泛型类别提供了可能性的可能性。为线性和非线性PDE提供了数值应用和讨论。
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.
