# Riemannian Perspective on Matrix Factorization

arXiv: https://arxiv.org/abs/2102.00937 · HTML: https://mlsanity.com/paper/2102.00937/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 46 references that are in the map.
- First posted on arXiv 1 Feb 2021.

## Builds on (most cited first, 14 of 14)

- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach](https://mlsanity.com/paper/1609.03240/) (2016)
- [Understanding Alternating Minimization for Matrix Completion](https://mlsanity.com/paper/1312.0925/) (2013)
- [Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems](https://mlsanity.com/paper/1411.1134/) (2014)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [Fast matrix completion without the condition number](https://mlsanity.com/paper/1407.4070/) (2014)
- [Subgradient Descent Learns Orthogonal Dictionaries](https://mlsanity.com/paper/1810.10702/) (2018)
- [Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization](https://mlsanity.com/paper/2012.15467/) (2020)
- [Maximizing robustness of point-set registration by leveraging non-convexity](https://mlsanity.com/paper/2004.08772/) (2020)

## Built on by (most cited first, 3 of 3)

- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization](https://mlsanity.com/paper/2110.12121/) (2021)
- [Riemannian Bilevel Optimization](https://mlsanity.com/paper/2405.15816/) (2024)

## Abstract (arXiv)

We study the non-convex matrix factorization approach to matrix completion via Riemannian geometry. Based on an optimization formulation over a Grassmannian manifold, we characterize the landscape based on the notion of principal angles between subspaces. For the fully observed case, our results show that there is a region in which the cost is geodesically convex, and outside of which all critical points are strictly saddle. We empirically study the partially observed case based on our findings.