## Abstract

Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. The aim is to find a low rank approximation for nonnegative matrix 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 a separability assumption, which requires that the columns of the factor matrix are some columns of the input matrix M. In this paper, we generalize the separability assumption based on 3-factor NMF (M = P1SP2), and require that S is a submatrix of the input matrix M. We refer to this NMF as a Coseparable NMF (CoS-NMF). In the paper, we discuss and study mathematical properties of CoS-NMF, and present its relationships with other matrix factorizations such as generalized separable NMF, tri-symNMF, biorthogonal trifactorization and CUR decomposition. An optimization method for CoS-NMF is proposed, and an alternating fast gradient method is employed to determine the rows and the columns of M for the submatrix S. Numerical experiments on synthetic data sets, document data sets, and facial data sets are conducted to verify the effectiveness of the proposed CoS-NMF model. By comparison with state-of-the-art methods, the CoS-NMF model performs very well in a coclustering task by finding useful features, and keeps a good approximation to the input data matrix as well.

Original language | English |
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Pages (from-to) | 1393-1420 |

Number of pages | 28 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 44 |

Issue number | 3 |

Early online date | 15 Sept 2023 |

DOIs | |

Publication status | Published - Sept 2023 |

## Scopus Subject Areas

- Analysis

## User-Defined Keywords

- nonnegative matrix factorization
- separability
- coseparable
- coclustering