Algorithms for Boolean Matrix Factorization using Integer Programming
This work addresses the challenge of efficiently approximating binary matrices with lower reconstruction errors, which is incremental as it builds on existing BMF methods with improved optimization techniques.
The paper tackles the NP-hard problem of Boolean matrix factorization (BMF) by proposing an alternating optimization strategy using integer programming to solve subproblems and combining multiple solutions optimally, resulting in algorithms that outperform state-of-the-art methods on medium-scale problems.
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean OR and Boolean AND operations to perform matrix products, which leads to lower reconstruction errors. BMF is an NP-hard problem. In this paper, we first propose an alternating optimization (AO) strategy that solves the subproblem in one factor matrix in BMF using an integer program (IP). We also provide two ways to initialize the factors within AO. Then, we show how several solutions of BMF can be combined optimally using another IP. This allows us to come up with a new algorithm: it generates several solutions using AO and then combines them in an optimal way. Experiments show that our algorithms (available on gitlab) outperform the state of the art on medium-scale problems.