Boolean matrix has been used to represent digital information in many fields,
including bank transaction, crime records, natural language processing,
protein-protein interaction, etc. Boolean matrix factorization (BMF) aims to
find an approximation of a binary matrix as the Boolean product of two low rank
Boolean matrices, which could generate vast amount of information for the
patterns of relationships between the features and samples. Inspired by binary
matrix permutation theories and geometric segmentation, we developed a fast and
efficient BMF approach called MEBF (Median Expansion for Boolean
Factorization). Overall, MEBF adopted a heuristic approach to locate binary
patterns presented as submatrices that are dense in 1's. At each iteration,
MEBF permutates the rows and columns such that the permutated matrix is
approximately Upper Triangular-Like (UTL) with so-called Simultaneous
Consecutive-ones Property (SC1P). The largest submatrix dense in 1 would lies
on the upper triangular area of the permutated matrix, and its location was
determined based on a geometric segmentation of a triangular. We compared MEBF
with other state of the art approaches on data scenarios with different
sparsity and noise levels. MEBF demonstrated superior performances in lower
reconstruction error, and higher computational efficiency, as well as more
accurate sparse patterns than popular methods such as ASSO, PANDA and MP. We
demonstrated the application of MEBF on both binary and non-binary data sets,
and revealed its further potential in knowledge retrieving and data denoising.

本研究开发了一种名为 MEBF 的快速、高效的布尔矩阵因式分解方法，采用启发式方法以定位 “密集的 1” 的子矩阵为目标，其性能表现优于 ASSO、PANDA 和 MP 等其他常用方法，同时在二进制和非二进制数据集上的应用表明其具有知识检索和数据去噪的潜在能力。