Improved Algorithms for Exact and Approximate Boolean Matrix Decomposition

TL;DR

This paper introduces new algorithms for Boolean matrix decomposition that improve efficiency and accuracy, supported by theoretical formulas and tested on real datasets, with applications in data mining.

Contribution

The paper presents an exact formula for a maximal Boolean matrix J and proposes two heuristic algorithms for approximate decomposition, enhancing performance and interpretability.

Findings

One of the algorithms performs as well or better than existing methods on benchmark datasets.

Algorithms are based on an exact mathematical formula, ensuring interpretability.

Proposed methods are fast and effective for both exact and approximate Boolean matrix decomposition.

Abstract

An arbitrary $m\times n$ Boolean matrix $M$ can be decomposed {\em exactly} as $M =U\circ V$, where $U$ (resp. $V$) is an $m\times k$ (resp. $k\times n$) Boolean matrix and $\circ$ denotes the Boolean matrix multiplication operator. We first prove an exact formula for the Boolean matrix $J$ such that $M =M\circ J^T$ holds, where $J$ is maximal in the sense that if any 0 element in $J$ is changed to a 1 then this equality no longer holds. Since minimizing $k$ is NP-hard, we propose two heuristic algorithms for finding suboptimal but good decomposition. We measure the performance (in minimizing $k$) of our algorithms on several real datasets in comparison with other representative heuristic algorithms for Boolean matrix decomposition (BMD). The results on some popular benchmark datasets demonstrate that one of our proposed algorithms performs as well or better on most of them. Our algorithms have a number of other advantages: They are based on exact mathematical formula, which can be interpreted intuitively. They can be used for approximation as well with competitive "coverage." Last but not least, they also run very fast. Due to interpretability issues in data mining, we impose the condition, called the "column use condition," that the columns of the factor matrix $U$ must form a subset of the columns of $M$.