# Guaranteed Matrix Completion via Non-convex Factorization

arXiv: https://arxiv.org/abs/1411.8003 · HTML: https://mlsanity.com/paper/1411.8003/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Nov 2014.

## Built on by (most cited first, 30 of 82)

- [Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorization](https://mlsanity.com/paper/2106.14289/) (2021)
- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [Implicit bias of SGD in L2-regularized linear DNNs: One-way jumps from high to low rank](https://mlsanity.com/paper/2305.16038/) (2023)
- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Weight decay induces low-rank attention layers](https://mlsanity.com/paper/2410.23819/) (2024)
- [Compressible Dynamics in Deep Overparameterized Low-Rank Learning & Adaptation](https://mlsanity.com/paper/2406.04112/) (2024)
- [Convergence of gradient descent for deep neural networks](https://mlsanity.com/paper/2203.16462/) (2022)
- [Improving Fairness for Data Valuation in Horizontal Federated Learning](https://mlsanity.com/paper/2109.09046/) (2021)
- [Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery](https://mlsanity.com/paper/2109.11154/) (2021)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification](https://mlsanity.com/paper/2206.03345/) (2022)
- [The Law of Parsimony in Gradient Descent for Learning Deep Linear Networks](https://mlsanity.com/paper/2306.01154/) (2023)
- [Fair and efficient contribution valuation for vertical federated learning](https://mlsanity.com/paper/2201.02658/) (2022)
- [Personalized PCA: Decoupling Shared and Unique Features](https://mlsanity.com/paper/2207.08041/) (2022)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [Submodular + Concave](https://mlsanity.com/paper/2106.04769/) (2021)
- [Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization](https://mlsanity.com/paper/2207.01789/) (2022)
- [Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling](https://mlsanity.com/paper/2208.09723/) (2022)
- [Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization](https://mlsanity.com/paper/2504.09708/) (2025)
- [Implicit Regularization in Matrix Sensing via Mirror Descent](https://mlsanity.com/paper/2105.13831/) (2021)
- [Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing](https://mlsanity.com/paper/2204.11516/) (2022)
- [Time Series Forecasting via Learning Convolutionally Low-Rank Models](https://mlsanity.com/paper/2104.11510/) (2021)
- [Multi-source Learning via Completion of Block-wise Overlapping Noisy Matrices](https://mlsanity.com/paper/2105.10360/) (2021)
- [GNMR: A provable one-line algorithm for low rank matrix recovery](https://mlsanity.com/paper/2106.12933/) (2021)
- [Private Alternating Least Squares: Practical Private Matrix Completion with Tighter Rates](https://mlsanity.com/paper/2107.09802/) (2021)
- [A Novel Stochastic Gradient Descent Algorithm for Learning Principal Subspaces](https://mlsanity.com/paper/2212.04025/) (2022)
- [Crowdsourcing via Annotator Co-occurrence Imputation and Provable Symmetric Nonnegative Matrix Factorization](https://mlsanity.com/paper/2106.07193/) (2021)
- [Accelerating SGD for Highly Ill-Conditioned Huge-Scale Online Matrix Completion](https://mlsanity.com/paper/2208.11246/) (2022)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)

## Abstract (arXiv)

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of this formulation. In this paper, we establish a theoretical guarantee for the factorization formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization formulation, and recover the true low-rank matrix. We study the local geometry of a properly regularized factorization formulation and prove that any stationary point in a certain local region is globally optimal. A major difference of our work from the existing results is that we do not need resampling in either the algorithm or its analysis. Compared to other works on nonconvex optimization, one extra difficulty lies in analyzing nonconvex constrained optimization when the constraint (or the corresponding regularizer) is not "consistent" with the gradient direction. One technical contribution is the perturbation analysis for non-symmetric matrix factorization.