# Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems

arXiv: https://arxiv.org/abs/1606.01316 · HTML: https://mlsanity.com/paper/1606.01316/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2016.

## Built on by (most cited first, 2 of 2)

- [Fast quantum state reconstruction via accelerated non-convex programming](https://mlsanity.com/paper/2104.07006/) (2021)
- [Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery](https://mlsanity.com/paper/2410.16826/) (2024)

## Abstract (arXiv)

We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-definite (PSD) constraints and specific matrix norm-constraints. Such criteria appear in quantum state tomography and phase retrieval applications. We show that non-convex projected gradient descent favors local linear convergence in the factored space. We build our theory on a novel descent lemma, that non-trivially extends recent results on the unconstrained problem. The resulting algorithm is Projected Factored Gradient Descent, abbreviated as ProjFGD, and shows superior performance compared to state of the art on quantum state tomography and sparse phase retrieval applications.