# CUR Decompositions, Approximations, and Perturbations

arXiv: https://arxiv.org/abs/1903.09698 · HTML: https://mlsanity.com/paper/1903.09698/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Mar 2019.

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

- [SECURA: Sigmoid-Enhanced CUR Decomposition with Uninterrupted Retention and Low-Rank Adaptation in Large Language Models](https://mlsanity.com/paper/2502.18168/) (2025)
- [A Unifying Family of Data-Adaptive Partitioning Algorithms](https://mlsanity.com/paper/2412.16713/) (2024)

## Abstract (arXiv)

This article discusses a useful tool in dimensionality reduction and low-rank matrix approximation called the CUR decomposition. Various viewpoints of this method in the literature are synergized and are compared and contrasted; included in this is a new characterization of exact CUR decompositions. A novel perturbation analysis is performed on CUR approximations of noisy versions of low-rank matrices, which compares them with the putative CUR decomposition of the underlying low-rank part. Additionally, we give new column and row sampling results which allow one to conclude that a CUR decomposition of a low-rank matrix is attained with high probability. We then illustrate the stability of these sampling methods under the perturbations studied before, and provide numerical illustrations of the methods and bounds discussed.