mlsanityData to 7 Oct 2026

Long Random Matrices and Tensor Unfolding

arXiv 2110.10210 ↗First posted 19 Oct 2021math.PR cs.LG stat.MLcited by 10

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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.

Key facts. Cited by 10 papers in the map (data through 7 Oct 2026). Builds on 19 of its 59 references that are in the map. First posted on arXiv 19 Oct 2021.

Builds on

19 of 59 references.

References · 19

  1. 2016Tensor Decomposition for Signal Processing and Machine Learningcited by 223
  2. 2014A statistical model for tensor PCAcited by 72
  3. 2017Tensor SVD: Statistical and Computational Limitscited by 52
  4. 2020An Optimal Statistical and Computational Framework for Generalized Tensor Estimationcited by 39
  5. 2015Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectorscited by 38
  6. 2015Tensor principal component analysis via sum-of-squares proofscited by 35
  7. 2017The landscape of the spiked tensor modelcited by 34
  8. 2020Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limitscited by 29
  9. 2016Statistical limits of spiked tensor modelscited by 27
  10. 2019Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Modelcited by 25
  11. 2016Tensor Methods and Recommender Systemscited by 24
  12. 2018Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitionscited by 23
7 more
  1. 2018Notes on computational-to-statistical gaps: predictions using statistical physicscited by 22
  2. 2020Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrievalcited by 18
  3. 2017Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squarescited by 17
  4. 2019How to iron out rough landscapes and get optimal performances: Averaged Gradient Descent and its application to tensor PCAcited by 14
  5. 2020Open Problem: Average-Case Hardness of Hypergraphic Planted Clique Detectioncited by 9
  6. 2016Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronizationcited by 6
  7. 2020A Sharp Blockwise Tensor Perturbation Bound for Orthogonal Iterationcited by 5

Built on by

Cited by year

Citing papers per year: 2021: 1; 2022: 1; 2023: 4; 2024: 3; 2025: 1.

10 papers.

  1. 2021When Random Tensors meet Random Matricescited by 10
  2. 2023Optimizing Orthogonalized Tensor Deflation via Random Tensor Theorycited by 4
  3. 2024A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximationcited by 3
  4. 2022On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysiscited by 1
  5. 2023A Nested Matrix-Tensor Model for Noisy Multi-view Clusteringcited by 1
  6. 2023On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysiscited by 0
  7. 2023Topological complexity of spiked random polynomials and finite-rank spherical integralscited by 0
  8. 2024Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Modelcited by 0
  9. 2024Permutation recovery of spikes in noisy high-dimensional tensor estimationcited by 0
  10. 2025Statistical Limits in Random Tensors with Multiple Correlated Spikescited by 0