# The Global Geometry of Centralized and Distributed Low-rank Matrix Recovery without Regularization

arXiv: https://arxiv.org/abs/2003.10981 · HTML: https://mlsanity.com/paper/2003.10981/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Mar 2020.

## Built on by (most cited first, 7 of 7)

- [GNMR: A provable one-line algorithm for low rank matrix recovery](https://mlsanity.com/paper/2106.12933/) (2021)
- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm](https://mlsanity.com/paper/2201.13052/) (2022)
- [Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing](https://mlsanity.com/paper/2101.05113/) (2021)
- [In-depth Analysis of Low-rank Matrix Factorisation in a Federated Setting](https://mlsanity.com/paper/2409.08771/) (2024)
- [Scaled Gradient Descent for Ill-Conditioned Low-Rank Matrix Recovery with Optimal Sampling Complexity](https://mlsanity.com/paper/2604.00060/) (2026)

## Abstract (arXiv)

Low-rank matrix recovery is a fundamental problem in signal processing and machine learning. A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UV^T, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry---with no spurious local minima and satisfying the so-called strict saddle property---ensuring convergence to a global minimum for many local-search algorithms. Such results hold whenever the original objective function is restricted strongly convex and smooth. However, most of these results actually consider a modified cost function that includes a balancing regularizer. While useful for deriving theory, this balancing regularizer does not appear to be necessary in practice. In this work, we close this theory-practice gap by proving that the unaltered factorized non-convex problem, without the balancing regularizer, also has similar benign global geometry. Moreover, we also extend our theoretical results to the field of distributed optimization.