The low-rank canonical polyadic tensor decomposition is useful in data analysis and can be computed by solving a sequence of overdetermined least squares subproblems. Motivated by consideration of sparse tensors, we propose sketching each subproblem using leverage scores to select a subset of the rows, with probabilistic guarantees on the solution accuracy. We randomly sample rows proportional to leverage score upper bounds that can be efficiently computed using the special Khatri--Rao subproblem structure inherent in tensor decomposition. Crucially, for a $(d+1)$-way tensor, the number of rows in the sketched system is $O(r^d/\epsilon)$ for a decomposition of rank $r$ and $\epsilon$-accuracy in the least squares solve, independent of both the size and the number of nonzeros in the tensor. Along the way, we provide a practical solution to the generic matrix sketching problem of sampling overabundance for high-leverage-score rows, proposing to include such rows deterministically and combine repeated samples in the sketched system; we conjecture that this can lead to improved theoretical bounds. Numerical results on real-world large-scale tensors show the method is significantly faster than deterministic methods at nearly the same level of accuracy.


  1. tensor decomposition
  3. canonical polyadic
  4. CP
  5. matrix sketching
  6. leverage score sampling
  7. randomized numerical linear algebra
  8. RandNLA

MSC codes

  1. 15A69

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Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Practical Leverage-Based Sampling for Low-Rank Tensor Decomposition

Authors: Brett W. Larsen and Tamara G. Kolda

File: EndToEndComplexity.pdf

Type: PDF

Contents: CP-ARLS-LEV End-to-End Complexity.

File: EnronRRF.pdf

Type: PDF

Contents: Enron Tensor with RRF Initialization.

File: UberFullRuns.pdf

Type: PDF

Contents: Detailed Runs on Uber Tensor.

File: RedditFactors.pdf

Type: PDF

Contents: Visualizations of Reddit Factors.


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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1488 - 1517
ISSN (online): 1095-7162


Submitted: 20 August 2021
Accepted: 6 June 2022
Published online: 30 August 2022


  1. tensor decomposition
  3. canonical polyadic
  4. CP
  5. matrix sketching
  6. leverage score sampling
  7. randomized numerical linear algebra
  8. RandNLA

MSC codes

  1. 15A69



Funding Information

U.S. Department of Energy https://doi.org/10.13039/100000015
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-FG02-97ER25308

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