# Accurate and fast matrix factorization for low-rank learning

arXiv: https://arxiv.org/abs/2104.10785 · HTML: https://mlsanity.com/paper/2104.10785/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 39 references that are in the map.
- First posted on arXiv 21 Apr 2021.

## Builds on (most cited first, 5 of 5)

- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Low-Rank Modeling and Its Applications in Image Analysis](https://mlsanity.com/paper/1401.3409/) (2014)
- [Robust PCA by Manifold Optimization](https://mlsanity.com/paper/1708.00257/) (2017)
- [R3MC: A Riemannian three-factor algorithm for low-rank matrix completion](https://mlsanity.com/paper/1306.2672/) (2013)
- [Fast estimation of approximate matrix ranks using spectral densities](https://mlsanity.com/paper/1608.05754/) (2016)

## Built on by (most cited first, 1 of 1)

- [Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations](https://mlsanity.com/paper/2605.02279/) (2026)

## Abstract (arXiv)

In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the accuracy of the desired singular values and also all of their corresponding singular vectors are essential. As a real application, we evaluate the performance of our methods on the problem of Riemannian similarity learning between two various image datasets of MNIST and USPS.