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01-03-2023

Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

Author: Erna Begović Kovač

Published in: Calcolo | Issue 1/2023

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Abstract

For a general third-order tensor \({{\mathcal {A}}}\in {\mathbb {R}}^{n\times n\times n}\) the paper studies two closely related problems, an SVD-like tensor decomposition and an (approximate) tensor diagonalization. We develop a Jacobi-type algorithm that works on \(2\times 2\times 2\) subtensors and, in each iteration, maximizes the sum of squares of its diagonal entries. We show how the rotation angles are calculated and prove convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on higher-order tensors.

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Kovač, E.B.: Finding the closest normal structured matrix. Linear Algebra Appl. 617, 49–77 (2021)MathSciNetCrossRefMATH

Kovač, E.B., Kressner, D.: Structure-preserving low multilinear rank approximation of antisymmetric tensors. SIAM J. Matrix Anal. Appl. 38(3), 967–983 (2017)MathSciNetCrossRefMATH

Comon, P.: Tensor diagonalization, a useful tool in signal processing. IFAC Proc. Vol. 27(8), 77–82 (1994)CrossRef

Comon, P., Jutten, C. (eds.): Handbook of Blind Source Separation. Academic Press, Oxford (2010)

De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)MathSciNetCrossRefMATH

De Lathauwer, L., De Moor, B., Vandewalle, J.: Independent component analysis and (simultaneous) third-order tensor diagonalization. IEEE Trans. Signal Process. 49, 2262–2271 (2001)CrossRef

Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31(4), 2029–2054 (2010)MathSciNetCrossRefMATH

Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer Series in Computational Mathematics 42 (2012)

Ishteva, M., Absil, P.-A., Van Dooren, P.: Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors. SIAM J. Matrix Anal. Appl. 34(2), 651–672 (2013)MathSciNetCrossRefMATH

10.

Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefMATH

11.

Landsberg, J.M.: Tensors: Geometry and Applications. American Mathematical Society, New York (2011)MATH

12.

Li, J., Usevich, K., Comon, P.: On approximate diagonalization of third order symmetric tensors by orthogonal transformations. Linear Algebra Appl. 576(1), 324–351 (2019)MathSciNetCrossRefMATH

13.

Li, J., Usevich, K., Comon, P.: Globally convergent Jacobi-type algorithms for simultaneous orthogonal symmetric tensor diagonalization. SIAM J. Matrix Anal. Appl. 39(1), 1–22 (2018)MathSciNetCrossRefMATH

14.

Moravitz Martin, C.D., Van Loan, C.F.: A Jacobi-type method for computing orthogonal tensor decompositions. SIAM J. Matrix Anal. Appl. 30(3), 1219–11232 (2008)MathSciNetCrossRefMATH

15.

Usevich, K., Li, J., Comon, P.: Approximate matrix and tensor diagonalization by unitary transformations: convergence of Jacobi-type algorithms. SIAM J. Optim. 30(4), 2998–3028 (2020)MathSciNetCrossRefMATH

16.

Vannieuwenhoven, N., Vandebril, R., Meerbergen, K.: A new truncation strategy for the higher-order singular value decomposition. SIAM J. Sci. Comput. 34(2), A1027–A1052 (2012)MathSciNetCrossRefMATH

Title: Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization
Author: Erna Begović Kovač
Publication date: 01-03-2023
Publisher: Springer International Publishing
Published in: Calcolo / Issue 1/2023
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-022-00498-x

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Convergence of a Jacobi-type method for the approximate orthogonal tensor diagonalization

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