Tensor rank reduction via coordinate flows

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.

arXiv comment41 pages, 19 figures

References in corpus (14)

• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• The Magnus expansion and some of its applications
• Active subspace methods in theory and practice: applications to kriging surfaces
• Efficient tensor completion for color image and video recovery: Low-rank tensor train
• Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm
• Time integration of tensor trains
• Generic Construction of Efficient Matrix Product Operators
• Spectral tensor-train decomposition
• Fermionic orbital optimisation in tensor network states
• Gradient Flows for Optimisation and Quantum Control: Foundations and Applications
• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs
• Dynamic tensor approximation of high-dimensional nonlinear PDEs
• Stability analysis of hierarchical tensor methods for time-dependent PDEs

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