Data to 7 Oct 2026

Low-rank Solutions of Linear Matrix Equations via Procrustes Flow

arXiv 1507.03566 ↗First posted 13 Jul 2015math.OCcited by 97

Not on the map. Why?

Abstract (arXiv)

In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n1 × n2 matrix of rank r when the number of measurements exceeds a constant times (n1+n2)r.

Key facts. Cited by 97 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Jul 2015.

Builds on

No reference list yet.

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 23; 2022: 19; 2023: 15; 2024: 13; 2025: 12; 2026: 15.

97 papers.

Sort:

By area:

2021Exploiting Shared Representations for Personalized Federated Learningcited by 537
2021The Low-Rank Simplicity Bias in Deep Networkscited by 150
2023Rethinking Model Ensemble in Transfer-based Adversarial Attackscited by 88
2021Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstructioncited by 77
2023Transformers as Support Vector Machinescited by 67
2022Statistical Learning Theory for Control: A Finite Sample Perspectivecited by 52
2021Large Learning Rate Tames Homogeneity: Convergence and Balancing Effectcited by 48
2021Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorizationcited by 44
2021Implicit Regularization in Tensor Factorizationcited by 43
2023Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensingcited by 40
2022Algorithmic Regularization in Model-free Overparametrized Asymmetric Matrix Factorizationcited by 34
2023The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensingcited by 32

38 more

2021On the computational and statistical complexity of over-parameterized matrix sensingcited by 31
2022Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certificationcited by 21
2023How Over-Parameterization Slows Down Gradient Descent in Matrix Sensing: The Curses of Symmetry and Initializationcited by 18
2021Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorizationcited by 15
2023A General Algorithm for Solving Rank-one Matrix Sensingcited by 15
2024On the Crucial Role of Initialization for Matrix Factorizationcited by 15
2021Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergencecited by 13
2022Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterizationcited by 13
2023Convergence of Alternating Gradient Descent for Matrix Factorizationcited by 12
2025Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorizationcited by 12
2021Implicit Regularization in Matrix Sensing via Mirror Descentcited by 11
2022Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensingcited by 11
2021GNMR: A provable one-line algorithm for low rank matrix recoverycited by 10
2024Implicit Regularization of Sharpness-Aware Minimization for Scale-Invariant Problemscited by 9
2022A Validation Approach to Over-parameterized Matrix and Image Recoverycited by 8
2024Guaranteed Nonconvex Factorization Approach for Tensor Train Recoverycited by 8
2021Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurementscited by 7
2022Efficient learning of hidden state LTI state space models of unknown ordercited by 7
2022Noisy Low-rank Matrix Optimization: Geometry of Local Minima and Convergence Ratecited by 7
2024Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descentcited by 7
2021Factorization Approach for Low-complexity Matrix Completion Problems: Exponential Number of Spurious Solutions and Failure of Gradient Methodscited by 6
2022Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completioncited by 6
2023Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterizationcited by 6
2024Learning Linear Dynamics from Bilinear Observationscited by 6
2021Fast quantum state reconstruction via accelerated non-convex programmingcited by 5
2023Deep ReLU Networks Have Surprisingly Simple Polytopescited by 5
2023Fast and Accurate Estimation of Low-Rank Matrices from Noisy Measurements via Preconditioned Non-Convex Gradient Descentcited by 5
2025Probe-Free Low-Rank Activation Interventioncited by 5
2025Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Regioncited by 5
2021Sparse GCA and Thresholded Gradient Descentcited by 4
2022Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithmcited by 4
2022Robust Matrix Completion with Heavy-tailed Noisecited by 4
2022Rank-1 Matrix Completion with Gradient Descent and Small Random Initializationcited by 4
2023Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priorscited by 4
2024Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recoverycited by 4
2025Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient Descentcited by 4
2025A Scalable Factorization Approach for High-Order Structured Tensor Recoverycited by 4
2026Training Dynamics of Softmax Self-Attention: Fast Global Convergence via Preconditioningcited by 4

50 most cited of 97 All 97 (JSON)