Differential-geometric Newton method for the best rank-(R (1), R (2), R (3)) approximation of tensors

Ishteva, Mariya;De Lathauwer, Lieven;Absil, Pierre-Antoine;Van Huffel, Sabine
(2009) Numerical Algorithms — Vol. 51, n° 2, p. 179-194 (2009)

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Authors
  • Ishteva, MariyaUCLouvain
    Author
  • De Lathauwer, LievenUCLouvain
    Author
  • Absil, Pierre-AntoineUCLouvain
    Author
  • Van Huffel, SabineUCLouvain
    Author
Abstract
An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R (1), R (2), R (3)) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324-1342, 2000). This kind of algorithms are useful for many problems.
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Ishteva, M., De Lathauwer, L., Absil, P.-A., & Van Huffel, S. (2009). Differential-geometric Newton method for the best rank-(R (1), R (2), R (3)) approximation of tensors. Numerical Algorithms, 51(2), 179-194. https://doi.org/10.1007/s11075-008-9251-2 (Original work published 2009)