# An equivalence between critical points for rank constraints versus low-rank factorizations

arXiv: https://arxiv.org/abs/1812.00404 · HTML: https://mlsanity.com/paper/1812.00404/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2018.

## Built on by (most cited first, 4 of 4)

- [Noisy Low-rank Matrix Optimization: Geometry of Local Minima and Convergence Rate](https://mlsanity.com/paper/2203.03899/) (2022)
- [Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements](https://mlsanity.com/paper/2105.08232/) (2021)
- [Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points](https://mlsanity.com/paper/2302.07828/) (2023)
- [On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization](https://mlsanity.com/paper/2110.12121/) (2021)

## Abstract (arXiv)

Two common approaches in low-rank optimization problems are either working directly with a rank constraint on the matrix variable, or optimizing over a low-rank factorization so that the rank constraint is implicitly ensured. In this paper, we study the natural connection between the rank-constrained and factorized approaches. We show that all second-order stationary points of the factorized objective function correspond to fixed points of projected gradient descent run on the original problem (where the projection step enforces the rank constraint). This result allows us to unify many existing optimization guarantees that have been proved specifically in either the rank-constrained or the factorized setting, and leads to new results for certain settings of the problem. We demonstrate application of our results to several concrete low-rank optimization problems arising in matrix inverse problems.