February 2024 When random tensors meet random matrices
Mohamed El Amine Seddik, Maxime Guillaud, Romain Couillet
Author Affiliations +
Ann. Appl. Probab. 34(1A): 203-248 (February 2024). DOI: 10.1214/23-AAP1962


Relying on random matrix theory (RMT), this paper studies asymmetric order-d spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of Lim (In Proc. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (2005) 129–132), we show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric blockwise random matrix, that is constructed from contractions of the studied tensor with the singular vectors associated to its best rank-1 approximation. Our approach allows the exact characterization of the almost sure asymptotic singular value and alignments of the corresponding singular vectors with the true spike components when nij=1dnjci(0,1) with ni’s the tensor dimensions. In contrast to other works that rely mostly on tools from statistical physics to study random tensors, our results rely solely on classical RMT tools such as Stein’s lemma. Finally, classical RMT results concerning spiked random matrices are recovered as a particular case.

Funding Statement

This work was supported by the MIAILargeDATA Chair at University Grenoble Alpes led by R. Couillet and the UGA-HUAWEI LarDist project led by M. Guillaud.


We would like to thank Henrique Goulart, Pierre Common and Gérard Ben-Aroud for valuable discussions on the topic of random tensors.

This work was undertaken while M. Seddik and M. Guillaud were with Huawei Technologies France.


Download Citation

Mohamed El Amine Seddik. Maxime Guillaud. Romain Couillet. "When random tensors meet random matrices." Ann. Appl. Probab. 34 (1A) 203 - 248, February 2024. https://doi.org/10.1214/23-AAP1962


Received: 1 January 2022; Revised: 1 November 2022; Published: February 2024
First available in Project Euclid: 28 January 2024

MathSciNet: MR4696276
zbMATH: 07829141
Digital Object Identifier: 10.1214/23-AAP1962

Primary: 60B20
Secondary: 15B52

Keywords: Random matrix theory , random tensor theory , spiked models

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.34 • No. 1A • February 2024
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