Abstract
Higher-order tensors have become popular in many areas of applied mathematics such as statistics, scientific computing, signal processing or machine learning, notably thanks to the many possible ways of decomposing a tensor. In this paper, we focus on the best approximation in the least-squares sense of a higher-order tensor by a block term decomposition. Using variable projection, we express the tensor approximation problem as a minimization of a cost function on a Cartesian product of Stiefel manifolds. The effect of variable projection on the Riemannian gradient algorithm is studied through numerical experiments.
This work was supported by (1) “Communauté française de Belgique - Actions de Recherche Concertées” (contract ARC 14/19-060), (2) Research Council KU Leuven: C1 project C16/15/059-nD, (3) F.W.O.: project G.0830.14N, G.0881.14N, (4) Fonds de la Recherche Scientifique – FNRS and the Fonds Wetenschappelijk Onderzoek – Vlaanderen under EOS Project no. 30468160 (SeLMA), (5) EU: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Advanced Grant: BIOTENSORS (no. 339804). This paper reflects only the authors’ views and the Union is not liable for any use that may be made of the contained information.
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Notes
- 1.
The minimizer is unique if and only if the matrix \(\mathbf {P}(\mathbf {U},\mathbf {V},\mathbf {W})\) has full column rank which is the case almost everywhere (with respect to the Lebesgue measure) since \(m \ge n\).
- 2.
The Matlab code that produced the results is available at https://sites.uclouvain.be/absil/2018.01.
- 3.
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Olikier, G., Absil, PA., De Lathauwer, L. (2018). Variable Projection Applied to Block Term Decomposition of Higher-Order Tensors. In: Deville, Y., Gannot, S., Mason, R., Plumbley, M., Ward, D. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2018. Lecture Notes in Computer Science(), vol 10891. Springer, Cham. https://doi.org/10.1007/978-3-319-93764-9_14
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