# Ellipsoidal Rounding for Nonnegative Matrix Factorization Under Noisy Separability

arXiv: https://arxiv.org/abs/1309.5701 · HTML: https://mlsanity.com/paper/1309.5701/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2013.

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

- [Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach](https://mlsanity.com/paper/2109.11135/) (2021)
- [Refinement of Hottopixx Method for Nonnegative Matrix Factorization Under Noisy Separability](https://mlsanity.com/paper/2109.02863/) (2021)
- [On the Robustness of the Successive Projection Algorithm](https://mlsanity.com/paper/2411.16195/) (2024)
- [Endmember Extraction from Hyperspectral Images Using Self-Dictionary Approach with Linear Programming](https://mlsanity.com/paper/2404.13098/) (2024)
- [Convex optimization over a probability simplex](https://mlsanity.com/paper/2305.09046/) (2023)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)
- [Hyperspectral Image Data Reduction for Endmember Extraction](https://mlsanity.com/paper/2512.10506/) (2025)

## Abstract (arXiv)

We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research interest of this paper is to find the vectors as close to the vertices as possible in a situation in which noise is added to the data points. Our algorithm is designed to capture the shape of the convex hull of data points by using its enclosing ellipsoid. We show that the algorithm has correctness and robustness properties from theoretical and practical perspectives; correctness here means that if the data points do not contain any noise, the algorithm can find the vertices of their convex hull; robustness means that if the data points contain noise, the algorithm can find the near-vertices. Finally, we apply the algorithm to document clustering, and report the experimental results.