Algorithms for Boolean Matrix Factorization using Integer Programming and Heuristics

2025-12-03

AI Summary

This work addresses large-scale Boolean Matrix Factorization (BMF), aiming to enhance interpretability and approximation accuracy for binary data. The proposed method employs an alternating optimization framework: each single-factor subproblem is solved exactly via integer programming, integrated with an optimal rank-one factor selection strategy; greedy initialization and local search heuristics accelerate convergence; and a lightweight C++-implemented Boolean data structure optimizes memory usage and computational efficiency. The approach natively supports missing-value handling. Extensive experiments on multiple real-world datasets demonstrate that the algorithm significantly outperforms state-of-the-art BMF methods in both reconstruction accuracy and runtime efficiency—particularly exhibiting strong robustness and superior performance in scenarios with missing entries.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the matrix product, which improves interpretability and reduces the approximation error. It is also used in role mining and computer vision. In this paper, we first propose algorithms for BMF that perform alternating optimization (AO) of the factor matrices, where each subproblem is solved via integer programming (IP). We then design different approaches to further enhance AO-based algorithms by selecting an optimal subset of rank-one factors from multiple runs. To address the scalability limits of IP-based methods, we introduce new greedy and local-search heuristics. We also construct a new C++ data structure for Boolean vectors and matrices that is significantly faster than existing ones and is of independent interest, allowing our heuristics to scale to large datasets. We illustrate the performance of all our proposed methods and compare them with the state of the art on various real datasets, both with and without missing data, including applications in topic modeling and imaging.

Problem

• Develop integer programming and heuristic algorithms for Boolean matrix factorization
• Address scalability issues of IP methods with greedy and local-search heuristics
• Create efficient C++ data structures to handle large Boolean datasets

Innovation

• Alternating optimization via integer programming
• Greedy and local-search heuristics for scalability
• Fast C++ data structure for Boolean operations