# Co-Separable Nonnegative Matrix Factorization

arXiv: https://arxiv.org/abs/2109.00749 · HTML: https://mlsanity.com/paper/2109.00749/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 41 references that are in the map.
- First posted on arXiv 2 Sep 2021.

## Builds on (most cited first, 11 of 11)

- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering](https://mlsanity.com/paper/1608.04290/) (2016)
- [Improving CUR Matrix Decomposition and the Nystr"om Approximation via Adaptive Sampling](https://mlsanity.com/paper/1303.4207/) (2013)
- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation](https://mlsanity.com/paper/1310.7529/) (2013)
- [Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization](https://mlsanity.com/paper/1302.4385/) (2013)
- [Hierarchical Clustering of Hyperspectral Images using Rank-Two Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.7441/) (2013)
- [Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related](https://mlsanity.com/paper/1409.4320/) (2014)
- [A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary](https://mlsanity.com/paper/1610.01349/) (2016)
- [Generalized Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/1905.12995/) (2019)
- [Off-diagonal Symmetric Nonnegative Matrix Factorization](https://mlsanity.com/paper/2003.04775/) (2020)

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

- [Separable Quaternion Matrix Factorization for Polarization Images](https://mlsanity.com/paper/2207.14039/) (2022)

## Abstract (arXiv)

Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.