# A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation

arXiv: https://arxiv.org/abs/2402.03169 · HTML: https://mlsanity.com/paper/2402.03169/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 89 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 19 of 19)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [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)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Hyperspectral Super-Resolution: A Coupled Tensor Factorization Approach](https://mlsanity.com/paper/1804.05307/) (2018)
- [Notes on computational-to-statistical gaps: predictions using statistical physics](https://mlsanity.com/paper/1803.11132/) (2018)
- [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)
- [When Random Tensors meet Random Matrices](https://mlsanity.com/paper/2112.12348/) (2021)
- [A Tensor Approach to Learning Mixed Membership Community Models](https://mlsanity.com/paper/1302.2684/) (2013)
- [Spectral norm of random tensors](https://mlsanity.com/paper/1407.1870/) (2014)
- [Characterizing Spatiotemporal Transcriptome of Human Brain via Low Rank Tensor Decomposition](https://mlsanity.com/paper/1702.07449/) (2017)
- [Statistical mechanics of low-rank tensor decomposition](https://mlsanity.com/paper/1810.10065/) (2018)
- [EigenEvent: An Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance](https://mlsanity.com/paper/1406.3496/) (2014)
- [On the non-detectability of spiked large random tensors](https://mlsanity.com/paper/1802.07093/) (2018)

## Built on by (most cited first, 3 of 3)

- [Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Model](https://mlsanity.com/paper/2402.10677/) (2024)
- [Performance of Rank-One Tensor Approximation on Incomplete Data](https://mlsanity.com/paper/2504.07818/) (2025)
- [High-Dimensional Partial Least Squares: Spectral Analysis and Fundamental Limitations](https://mlsanity.com/paper/2512.15684/) (2025)

## Abstract (arXiv)

This work presents a comprehensive understanding of the estimation of a planted low-rank signal from a general spiked tensor model near the computational threshold. Relying on standard tools from the theory of large random matrices, we characterize the large-dimensional spectral behavior of the unfoldings of the data tensor and exhibit relevant signal-to-noise ratios governing the detectability of the principal directions of the signal. These results allow to accurately predict the reconstruction performance of truncated multilinear SVD (MLSVD) in the non-trivial regime. This is particularly important since it serves as an initialization of the higher-order orthogonal iteration (HOOI) scheme, whose convergence to the best low-multilinear-rank approximation depends entirely on its initialization. We give a sufficient condition for the convergence of HOOI and show that the number of iterations before convergence tends to 1 in the large-dimensional limit.