### Abstract

Mining discrete patterns in binary data is important for sub- sampling, compression, and clustering. We consider rank- one binary matrix approximations that identify the dominant patterns of the data, while preserving its discrete property. A best approximation on such data has a minimum set of inconsistent entries, i.e., mismatches between the given binary data and the approximate matrix. Due to the hardness of the problem, previous accounts of such problems employ heuristics and the resulting approximation may be far away from the optimal one. In this paper, we show that the rank-one binary matrix approximation can be reformulated as a 0-1 integer linear program (ILP). However, the ILP formulation is computationally expensive even for small-size matrices. We propose a linear program (LP) relaxation, which is shown to achieve a guaranteed approximation error bound. We further extend the proposed formulations using the regularization technique, which is commonly employed to address overfitting. The LP formulation is restricted to medium-size matrices, due to the large number of variables involved for large matrices. Interestingly, we show that the proposed approximate formulation can be transformed into an instance of the minimum s-t cut problem, which can be solved efficiently by finding maximum flows. Our empirical study shows the efficiency of the proposed algorithm based on the maximum flow. Results also confirm the established theoretical bounds.

Original language | English (US) |
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Title of host publication | KDD '09 |

Subtitle of host publication | Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |

Pages | 757-765 |

Number of pages | 9 |

DOIs | |

State | Published - Nov 9 2009 |

Event | 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '09 - Paris, France Duration: Jun 28 2009 → Jul 1 2009 |

### Publication series

Name | Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |
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### Conference

Conference | 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '09 |
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Country | France |

City | Paris |

Period | 6/28/09 → 7/1/09 |

### Keywords

- Binary matrix factorization
- Integer linear program
- Maximum flow
- Minimum cut
- Rank-one
- Regularization

### ASJC Scopus subject areas

- Software
- Information Systems

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## Cite this

*KDD '09: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*(pp. 757-765). (Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining). https://doi.org/10.1145/1557019.1557103