# Rank-one matrix estimation: analytic time evolution of gradient descent dynamics

arXiv: https://arxiv.org/abs/2105.12257 · HTML: https://mlsanity.com/paper/2105.12257/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 25 May 2021.

## Builds on (most cited first, 11 of 11)

- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach](https://mlsanity.com/paper/1609.03240/) (2016)
- [Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models](https://mlsanity.com/paper/1902.00139/) (2019)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems](https://mlsanity.com/paper/1411.1134/) (2014)
- [Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model](https://mlsanity.com/paper/1907.08226/) (2019)
- [Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference](https://mlsanity.com/paper/1812.09066/) (2018)
- [On the low-rank approach for semidefinite programs arising in synchronization and community detection](https://mlsanity.com/paper/1602.04426/) (2016)
- [On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization](https://mlsanity.com/paper/1809.11083/) (2018)

## Built on by (most cited first, 5 of 5)

- [Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature model](https://mlsanity.com/paper/2110.11805/) (2021)
- [Statistical limits of dictionary learning: random matrix theory and the spectral replica method](https://mlsanity.com/paper/2109.06610/) (2021)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models](https://mlsanity.com/paper/2204.04476/) (2022)
- [Gradient flow on extensive-rank positive semi-definite matrix denoising](https://mlsanity.com/paper/2303.09474/) (2023)

## Abstract (arXiv)

We consider a rank-one symmetric matrix corrupted by additive noise. The rank-one matrix is formed by an n-component unknown vector on the sphere of radius √n, and we consider the problem of estimating this vector from the corrupted matrix in the high dimensional limit of n large, by gradient descent for a quadratic cost function on the sphere. Explicit formulas for the whole time evolution of the overlap between the estimator and unknown vector, as well as the cost, are rigorously derived. In the long time limit we recover the well known spectral phase transition, as a function of the signal-to-noise ratio. The explicit formulas also allow to point out interesting transient features of the time evolution. Our analysis technique is based on recent progress in random matrix theory and uses local versions of the semi-circle law.