# A Random Matrix Perspective on Random Tensors

arXiv: https://arxiv.org/abs/2108.00774 · HTML: https://mlsanity.com/paper/2108.00774/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 2 Aug 2021.

## Builds on (most cited first, 13 of 13)

- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Tensor principal component analysis via sum-of-squares proofs](https://mlsanity.com/paper/1507.03269/) (2015)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [Provable Tensor Factorization with Missing Data](https://mlsanity.com/paper/1406.2784/) (2014)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [The Dynamics of Learning: A Random Matrix Approach](https://mlsanity.com/paper/1805.11917/) (2018)
- [A random matrix analysis and improvement of semi-supervised learning for large dimensional data](https://mlsanity.com/paper/1711.03404/) (2017)
- [Consistent Semi-Supervised Graph Regularization for High Dimensional Data](https://mlsanity.com/paper/2006.07575/) (2020)
- [A Large Dimensional Analysis of Least Squares Support Vector Machines](https://mlsanity.com/paper/1701.02967/) (2017)
- [On the non-detectability of spiked large random tensors](https://mlsanity.com/paper/1802.07093/) (2018)

## Built on by (most cited first, 12 of 12)

- [Data-driven emergence of convolutional structure in neural networks](https://mlsanity.com/paper/2202.00565/) (2022)
- [When Random Tensors meet Random Matrices](https://mlsanity.com/paper/2112.12348/) (2021)
- [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)
- [Permutation Equivariant Neural Networks for Symmetric Tensors](https://mlsanity.com/paper/2503.11276/) (2025)

## Abstract (arXiv)

Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits of estimators of that signal inevitably calls for the study of random tensors. Substantial progress has been recently achieved on this subject in the large-dimensional limit. Yet, some of the most significant among these results--in particular, a precise characterization of the abrupt phase transition (with respect to signal-to-noise ratio) that governs the performance of the maximum likelihood (ML) estimator of a symmetric rank-one model with Gaussian noise--were derived based of mean-field spin glass theory, which is not easily accessible to non-experts. In this work, we develop a sharply distinct and more elementary approach, relying on standard but powerful tools brought by years of advances in random matrix theory. The key idea is to study the spectra of random matrices arising from contractions of a given random tensor. We show how this gives access to spectral properties of the random tensor itself. For the aforementioned rank-one model, our technique yields a hitherto unknown fixed-point equation whose solution precisely matches the asymptotic performance of the ML estimator above the phase transition threshold in the third-order case. A numerical verification provides evidence that the same holds for orders 4 and 5, leading us to conjecture that, for any order, our fixed-point equation is equivalent to the known characterization of the ML estimation performance that had been obtained by relying on spin glasses. Moreover, our approach sheds light on certain properties of the ML problem landscape in large dimensions and can be extended to other models, such as asymmetric and non-Gaussian.