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**A practical randomized CP tensor decomposition.**
*(English)*
Zbl 1444.65016

Authors’ abstract: The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.

Reviewer: Gunther Schmidt (München)

### MSC:

65F99 | Numerical linear algebra |

15A69 | Multilinear algebra, tensor calculus |

68W20 | Randomized algorithms |

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\textit{C. Battaglino} et al., SIAM J. Matrix Anal. Appl. 39, No. 2, 876--901 (2018; Zbl 1444.65016)

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