# Matrix Completion has No Spurious Local Minimum

arXiv: https://arxiv.org/abs/1605.07272 · HTML: https://mlsanity.com/paper/1605.07272/

## Key facts

- Cited by 125 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2016.

## Built on by (most cited first, 30 of 125)

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- [On the Optimization Landscape of Neural Collapse under MSE Loss: Global Optimality with Unconstrained Features](https://mlsanity.com/paper/2203.01238/) (2022)
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- [An Unconstrained Layer-Peeled Perspective on Neural Collapse](https://mlsanity.com/paper/2110.02796/) (2021)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Are All Losses Created Equal: A Neural Collapse Perspective](https://mlsanity.com/paper/2210.02192/) (2022)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [LoRA Training in the NTK Regime has No Spurious Local Minima](https://mlsanity.com/paper/2402.11867/) (2024)
- [Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscape](https://mlsanity.com/paper/2402.01258/) (2024)
- [Global Convergence of Gradient Descent for Asymmetric Low-Rank Matrix Factorization](https://mlsanity.com/paper/2106.14289/) (2021)
- [Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2209.05045/) (2022)
- [Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing](https://mlsanity.com/paper/2301.11500/) (2023)
- [Provable Model-based Nonlinear Bandit and Reinforcement Learning: Shelve Optimism, Embrace Virtual Curvature](https://mlsanity.com/paper/2102.04168/) (2021)
- [Faster Rates of Convergence to Stationary Points in Differentially Private Optimization](https://mlsanity.com/paper/2206.00846/) (2022)
- [On the computational and statistical complexity of over-parameterized matrix sensing](https://mlsanity.com/paper/2102.02756/) (2021)
- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [Compressible Dynamics in Deep Overparameterized Low-Rank Learning & Adaptation](https://mlsanity.com/paper/2406.04112/) (2024)
- [Stochastic Continuous Submodular Maximization: Boosting via Non-oblivious Function](https://mlsanity.com/paper/2201.00703/) (2022)
- [Global Convergence of Sub-gradient Method for Robust Matrix Recovery: Small Initialization, Noisy Measurements, and Over-parameterization](https://mlsanity.com/paper/2202.08788/) (2022)
- [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)
- [Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles](https://mlsanity.com/paper/2306.14853/) (2023)
- [Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization](https://mlsanity.com/paper/2102.12430/) (2021)
- [Clustering a Mixture of Gaussians with Unknown Covariance](https://mlsanity.com/paper/2110.01602/) (2021)
- [Identifying good directions to escape the NTK regime and efficiently learn low-degree plus sparse polynomials](https://mlsanity.com/paper/2206.03688/) (2022)
- [Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization](https://mlsanity.com/paper/2207.01789/) (2022)
- [Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2110.04814/) (2021)
- [Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization](https://mlsanity.com/paper/2504.09708/) (2025)
- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [Escape saddle points by a simple gradient-descent based algorithm](https://mlsanity.com/paper/2111.14069/) (2021)

## Abstract (arXiv)

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for textitpositive semidefinite matrix completion has no spurious local minima --- all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve positive semidefinite matrix completion with textitarbitrary initialization in polynomial time. The result can be generalized to the setting when the observed entries contain noise. We believe that our main proof strategy can be useful for understanding geometric properties of other statistical problems involving partial or noisy observations.