Robust Low-Rank Tensor Decomposition with the L₂ Criterion

Abstract

The growing prevalence of tensor data, or multiway arrays, in science and engineering applications motivates the need for tensor decompositions that are robust against outliers. In this article, we present a robust Tucker decomposition estimator based on the L₂ criterion, called the Tucker-${\mathrm{L}}_2\mathrm{E}$. Our numerical experiments demonstrate that Tucker-${\mathrm{L}}_2\mathrm{E}$ has empirically stronger recovery performance in more challenging high-rank scenarios compared with existing alternatives. The appropriate Tucker-rank can be selected in a data-driven manner with cross-validation or hold-out validation. The practical effectiveness of Tucker-${\mathrm{L}}_2\mathrm{E}$ is validated on real data applications in fMRI tensor denoising, PARAFAC analysis of fluorescence data, and feature extraction for classification of corrupted images.

KEYWORDS:

• Inverse problem
• ${\mathrm{L}}_2$ criterion
• Nonconvexity
• Robustness
• Tucker decomposition

Supplementary Materials

Supplement: A pdf file that contains derivation of gradient, an alternative initialization strategy named spectral initialization with diagonal deletion, proof of Theorem 4.1, details of parameter choices, and a run time comparison with the baseline methods.

Software: Matlab code of the described method, along with scripts to reproduce some of the figures in Sections 5 and 6.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

We are grateful to the associate editor, editor, and anonymous referee for their valuable comments and suggestions, which greatly improved the presentation of this article. We thank Haixu Ma and Xiaoqian Liu for their assistance in testing the software.

Notes

1 https://www.tensortoolbox.org/

2 https://github.com/stephenbeckr/L-BFGS-B-C

3 https://www.osirix-viewer.com

4 http://www.models.life.ku.dk/dorrit

Additional information

Funding

This research was partially funded by grants from the National Institute of General Medical Sciences (R01GM135928: EC, R01GM126550: YL) and the National Science Foundation (DMS-2201136: EC, DMS-2100729: YL).