# Long Random Matrices and Tensor Unfolding

arXiv: https://arxiv.org/abs/2110.10210 · HTML: https://mlsanity.com/paper/2110.10210/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 59 references that are in the map.
- First posted on arXiv 19 Oct 2021.

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

- [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 SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors](https://mlsanity.com/paper/1512.02337/) (2015)
- [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)
- [Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limits](https://mlsanity.com/paper/2005.10743/) (2020)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model](https://mlsanity.com/paper/1907.08226/) (2019)
- [Tensor Methods and Recommender Systems](https://mlsanity.com/paper/1603.06038/) (2016)
- [Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions](https://mlsanity.com/paper/1804.02686/) (2018)
- [Notes on computational-to-statistical gaps: predictions using statistical physics](https://mlsanity.com/paper/1803.11132/) (2018)
- [Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval](https://mlsanity.com/paper/2006.06997/) (2020)
- [Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares](https://mlsanity.com/paper/1705.02973/) (2017)
- [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)
- [Open Problem: Average-Case Hardness of Hypergraphic Planted Clique Detection](https://mlsanity.com/paper/2009.05870/) (2020)
- [Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization](https://mlsanity.com/paper/1609.05573/) (2016)
- [A Sharp Blockwise Tensor Perturbation Bound for Orthogonal Iteration](https://mlsanity.com/paper/2008.02437/) (2020)

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

- [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)
- [A Nested Matrix-Tensor Model for Noisy Multi-view Clustering](https://mlsanity.com/paper/2305.19992/) (2023)
- [On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2310.18717/) (2023)
- [Topological complexity of spiked random polynomials and finite-rank spherical integrals](https://mlsanity.com/paper/2312.12323/) (2023)
- [Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Model](https://mlsanity.com/paper/2402.10677/) (2024)
- [Permutation recovery of spikes in noisy high-dimensional tensor estimation](https://mlsanity.com/paper/2412.14650/) (2024)
- [Statistical Limits in Random Tensors with Multiple Correlated Spikes](https://mlsanity.com/paper/2503.03356/) (2025)

## Abstract (arXiv)

In this paper, we consider the singular values and singular vectors of low rank perturbations of large rectangular random matrices, in the regime the matrix is "long": we allow the number of rows (columns) to grow polynomially in the number of columns (rows). We prove there exists a critical signal-to-noise ratio (depending on the dimensions of the matrix), and the extreme singular values and singular vectors exhibit a BBP type phase transition. As a main application, we investigate the tensor unfolding algorithm for the asymmetric rank-one spiked tensor model, and obtain an exact threshold, which is independent of the procedure of tensor unfolding. If the signal-to-noise ratio is above the threshold, tensor unfolding detects the signals; otherwise, it fails to capture the signals.