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Optimal Matrix-Mimetic Tensor Algebras via Variable Projection.

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Titel: Optimal Matrix-Mimetic Tensor Algebras via Variable Projection.
Autoren: Newman, Elizabeth1,2 (AUTHOR) e.newman@tufts.edu, Keegan, Katherine1 (AUTHOR) katherine.emiri.keegan@emory.edu
Quelle: SIAM Journal on Matrix Analysis & Applications. 2025, Vol. 46 Issue 3, p1764-1790. 27p.
Schlagwörter: *MATHEMATICAL optimization, *QUANTITATIVE research, TENSOR algebra, LINEAR operators, MATHEMATICAL transformations, REPRODUCIBLE research, CALCULUS of tensors
Abstract: Recent advances in matrix-mimetic tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogously to matrices. Underlying the tensor operation is an algebraic framework parameterized by an invertible linear transformation. The choice of linear mapping is crucial to representation quality and, in practice, is made heuristically based on expected correlations in the data. However, in many cases, these correlations are unknown and common heuristics lead to suboptimal performance. In this work, we simultaneously learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. Our new framework explicitly captures the coupling between the transformation and representation using variable projection. We preserve the invertibility of the linear mapping by learning orthogonal transformations with Riemannian optimization. We provide an original theory of the uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm. We demonstrate the generality of our framework through numerical experiments on a wide range of applications, including financial index tracking, image compression, and reduced order modeling. We have published all the code related to this work at . Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [3.64MB]). [ABSTRACT FROM AUTHOR]
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Datenbank: Business Source Index
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Abstract:Recent advances in matrix-mimetic tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises from interpreting tensors as operators that can be multiplied, factorized, and analyzed analogously to matrices. Underlying the tensor operation is an algebraic framework parameterized by an invertible linear transformation. The choice of linear mapping is crucial to representation quality and, in practice, is made heuristically based on expected correlations in the data. However, in many cases, these correlations are unknown and common heuristics lead to suboptimal performance. In this work, we simultaneously learn optimal linear mappings and corresponding tensor representations without relying on prior knowledge of the data. Our new framework explicitly captures the coupling between the transformation and representation using variable projection. We preserve the invertibility of the linear mapping by learning orthogonal transformations with Riemannian optimization. We provide an original theory of the uniqueness of the transformation and convergence analysis of our variable-projection-based algorithm. We demonstrate the generality of our framework through numerical experiments on a wide range of applications, including financial index tracking, image compression, and reduced order modeling. We have published all the code related to this work at . Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at and in the supplementary materials ( [3.64MB]). [ABSTRACT FROM AUTHOR]
ISSN:08954798
DOI:10.1137/24M1702635