# When Random Tensors meet Random Matrices

arXiv: https://arxiv.org/abs/2112.12348 · HTML: https://mlsanity.com/paper/2112.12348/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 23 Dec 2021.

## Builds on (most cited first, 8 of 8)

- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Introduction to Tensor Decompositions and their Applications in Machine Learning](https://mlsanity.com/paper/1711.10781/) (2017)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [How to iron out rough landscapes and get optimal performances: Averaged Gradient Descent and its application to tensor PCA](https://mlsanity.com/paper/1905.12294/) (2019)
- [A Random Matrix Perspective on Random Tensors](https://mlsanity.com/paper/2108.00774/) (2021)
- [Long Random Matrices and Tensor Unfolding](https://mlsanity.com/paper/2110.10210/) (2021)

## Built on by (most cited first, 10 of 10)

- [Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory](https://mlsanity.com/paper/2302.05798/) (2023)
- [A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation](https://mlsanity.com/paper/2402.03169/) (2024)
- [On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2211.09004/) (2022)
- [Hotelling Deflation on Large Symmetric Spiked Tensors](https://mlsanity.com/paper/2304.10248/) (2023)
- [A Nested Matrix-Tensor Model for Noisy Multi-view Clustering](https://mlsanity.com/paper/2305.19992/) (2023)
- [Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise](https://mlsanity.com/paper/2302.03306/) (2023)
- [On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2310.18717/) (2023)
- [Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Model](https://mlsanity.com/paper/2402.10677/) (2024)
- [Statistical Limits in Random Tensors with Multiple Correlated Spikes](https://mlsanity.com/paper/2503.03356/) (2025)
- [Performance of Rank-One Tensor Approximation on Incomplete Data](https://mlsanity.com/paper/2504.07818/) (2025)

## Abstract (arXiv)

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, 2005), we show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric textitblock-wise random matrix, that is constructed from textitcontractions 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 fracnisumj=1d nj→ ci∈ (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.