# Global Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation

arXiv: https://arxiv.org/abs/1506.07405 · HTML: https://mlsanity.com/paper/1506.07405/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2015.

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

- [SubTrack++ : Gradient Subspace Tracking for Scalable LLM Training](https://mlsanity.com/paper/2502.01586/) (2025)
- [Mining of Switching Sparse Networks for Missing Value Imputation in Multivariate Time Series](https://mlsanity.com/paper/2409.09930/) (2024)
- [Geometrically Principled Randomized Optimization for Efficient LLM Training](https://mlsanity.com/paper/2510.01878/) (2025)
- [Local Linear Convergence of Gradient Methods for Subspace Optimization via Strict Complementarity](https://mlsanity.com/paper/2202.04020/) (2022)
- [Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation](https://mlsanity.com/paper/2505.10882/) (2025)

## Abstract (arXiv)

It has been observed in a variety of contexts that gradient descent methods have great success in solving low-rank matrix factorization problems, despite the relevant problem formulation being non-convex. We tackle a particular instance of this scenario, where we seek the d-dimensional subspace spanned by a streaming data matrix. We apply the natural first order incremental gradient descent method, constraining the gradient method to the Grassmannian. In this paper, we propose an adaptive step size scheme that is greedy for the noiseless case, that maximizes the improvement of our metric of convergence at each data index t, and yields an expected improvement for the noisy case. We show that, with noise-free data, this method converges from any random initialization to the global minimum of the problem. For noisy data, we provide the expected convergence rate of the proposed algorithm per iteration.