Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.

Addressing the interpretability problem of NMF on Boolean data, Boolean Matrix Factorization (BMF) uses Boolean algebra to decompose the input into low-rank Boolean factor matrices. These matrices are highly interpretable and very useful in practice, but they come at the high computational cost of solving an NP-hard combinatorial optimization problem. To reduce the computational burden, we propose to relax BMF continuously using a novel elastic-binary regularizer, from which we derive a proximal gradient algorithm. Through an extensive set of experiments, we demonstrate that our method works well in practice: On synthetic data, we show that it converges quickly, recovers the ground truth precisely, and estimates the simulated rank exactly. On real-world data, we improve upon the state of the art in recall, loss, and runtime, and a case study from the medical domain confirms that our results are easily interpretable and semantically meaningful.

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.

Pattern set mining, which is the task of finding a good set of patterns instead of all patterns, is a fundamental problem in data mining. Many different definitions of what constitutes a good set have been proposed in recent years. In this paper, we consider the reconstruction error as a proxy measure for the goodness of the set, and concentrate on the adjacent problem of how to find a good set efficiently. We propose a method based on bottom-k hashing for efficiently selecting the set and extend the method for the common case where the patterns might only appear in approximate form in the data. Our approach has applications in tiling databases, Boolean matrix factorization, and redescription mining, among others. We show that our hashing-based approach is significantly faster than the standard greedy algorithm while obtaining almost equally good results in both synthetic and real-world data sets.

Addressing the interpretability problem of NMF on Boolean data, Boolean Matrix Factorization (BMF) uses Boolean algebra to decompose the input into low-rank Boolean factor matrices. These matrices are highly interpretable and very useful in practice, but they come at the high computational cost of solving an NP-hard combinatorial optimization problem. To reduce the computational burden, we propose to relax BMF continuously using a novel elastic-binary regularizer, from which we derive a proximal gradient algorithm. Through an extensive set of experiments, we demonstrate that our method works well in practice: On synthetic data, we show that it converges quickly, recovers the ground truth precisely, and estimates the simulated rank exactly. On real-world data, we improve upon the state of the art in recall, loss, and runtime, and a case study from the medical domain confirms that our results are easily interpretable and semantically meaningful.

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.

Boolean matrix factorization (BMF) is a variant of the standard matrix factorization problem in the Boolean semiring: given a binary matrix, the task is to find two smaller binary matrices so that their product, taken over the Boolean semiring, is as close to the original matrix as possible. Because the matrix product is not done over a field but over a semiring, many standard matrix factorization techniques fail to work. Indeed, finding the best Boolean factorization is computationally hard. The computational hardness of the problem has not prevented people from studying it. In psychometrics, some of the first algorithms appeared in the 1980's (see Bělohlávek and Trnecka (2018)). Even before that, mathematicians studying combinatorics had studied the "Boolean linear algebra" (Kim, 1982; Monson et al., 1995).

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.

Probabilistic approach to Boolean matrix factorization can provide solutions robust against noise and missing values with linear computational complexity. However, the assumption about latent factors can be problematic in real world applications. This study proposed a new probabilistic algorithm free of assumptions of latent factors, while retaining the advantages of previous algorithms. Real data experiment showed that our algorithm was favourably compared with current state-of-the-art probabilistic algorithms.