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Tensor rank reduction via coordinate flows.
- Source :
-
Journal of Computational Physics . Oct2023, Vol. 491, pN.PAG-N.PAG. 1p. - Publication Year :
- 2023
-
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. • Riemannian optimization of tensor ridge functions. • Tensor rank reduction via coordinate flows. • PDEs in curvilinear coordinate systems. • Rank-adaptive tensor methods. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00219991
- Volume :
- 491
- Database :
- Academic Search Index
- Journal :
- Journal of Computational Physics
- Publication Type :
- Academic Journal
- Accession number :
- 170067906
- Full Text :
- https://doi.org/10.1016/j.jcp.2023.112378