Pencil-based algorithms for tensor rank decomposition are not stable

Carlos Beltrán, Paul Breiding, and Nick Vannieuwenhoven

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Submission date: 12. Jul. 2018
published in: SIAM journal on matrix analysis and applications, 40 (2019) 2, p. 739-773 
DOI number (of the published article): 10.1137/18M1200531
Bibtex
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We prove the existence of an open set of n₁ × n₂ × n₃ tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, is arbitrarily numerically forward unstable. Our analysis shows that this problem is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n₁ × n₂ × 2 tensors than for the n₁ × n₂ × n₃ input tensor. Moreover, we present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits.