51. Tensor rank reduction via coordinate flows.
- Author
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Dektor, Alec and Venturi, Daniele
- Subjects
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CURVILINEAR coordinates , *COORDINATE transformations , *APPROXIMATION algorithms , *LINEAR statistical models - 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]
- Published
- 2023
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