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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
30/2019

The average condition number of most tensor rank decomposition problems is infinite

Carlos Beltrán, Paul Breiding and Nick Vannieuwenhoven

Abstract

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling.

We show for random rank-2 tensors with Gaussian density that the expected value of the condition number is infinite. Under some mild additional assumption, we show that the same is true for most higher ranks $r\geq 3$ as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 Gaussian tensors have finite expected angular condition number.

Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms that compute the CPD. Finally, we supply numerical experiments.

Received:
18.03.19
Published:
19.03.19

Related publications

inJournal
2023 Journal Open Access
Carlos Beltrán, Paul Breiding and Nick Vannieuwenhoven

The average condition number of most tensor rank decomposition problems is infinite

In: Foundations of computational mathematics, 23 (2023) 2, pp. 433-491