The goal of Boolean Matrix Factorization (BMF) is to approximate a given binary matrix as the product of two low-rank binary factor matrices, where the product of the factor matrices is computed under the Boolean algebra. While the problem is computationally hard, it is also attractive because the binary nature of the factor matrices makes them highly interpretable. In the last decade, BMF has received a considerable amount of attention in the data mining and formal concept analysis communities and, more recently, the machine learning and the theory communities also started studying BMF. In this survey, we give a concise summary of the efforts of all of these communities and raise some open questions which in our opinion require further investigation.
The Boolean matrix factorization problem consists in approximating a matrix by the Boolean product of two smaller Boolean matrices. To obtain optimal solutions when the matrices to be factorized are small, we propose SAT and MaxSAT encoding; however, when the matrices to be factorized are large, we propose a heuristic based on the search for maximal biclique edge cover. We experimentally demonstrate that our approaches allow a better factorization than existing approaches while keeping reasonable computation times. Our methods also allow the handling of incomplete matrices with missing entries…
Probabilistic approach to Boolean matrix factorization can provide solutions robustagainst 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 latentfactors, while retaining the advantages of previous algorithms. Real data experimentshowed that our algorithm was favourably compared with current state-of-the-artprobabilistic algorithms.
Given labeled data represented by a binary matrix, we consider the task to derive a Boolean matrix factorization which identifies commonalities and specifications among the classes. While existing works focus on rank-one factorizations which are either specific or common to the classes, we derive class-specific alterations from common factorizations as well. Therewith, we broaden the applicability of our new method to datasets whose class-dependencies have a more complex structure. On the basis of synthetic and real-world datasets, we show on the one hand that our method is able to filter stru…
National Research University Higher School of Economics（国家研究型高等经济大学）；LORIA（洛林计算机科学及其应用实验室）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；University of Lorraine（洛林大学）；Palacký University Olomouc（奥洛穆茨帕拉茨基大学）
During the past few years Boolean matrix factorization (BMF) has become an important direction in data analysis. The minimum description length principle (MDL) was successfully adapted in BMF for the model order selection. Nevertheless, a BMF algorithm performing good results from the standpoint of standard measures in BMF is missing. In this paper, we propose a novel from-below Boolean matrix factorization algorithm based on formal concept analysis. The algorithm utilizes the MDL principle as a criterion for the factor selection. On various experiments we show that the proposed algorithm outp…
Boolean matrix factorization (BMF) is a popular and powerful technique for inferring knowledge from data. The mining result is the Boolean product of two matrices, approximating the input dataset. The Boolean product is a disjunction of rank-1 binary matrices, each describing a feature-relation, called pattern, for a group of samples. Yet, there are no guarantees that any of the returned patterns do not actually arise from noise, i.e., are false discoveries. In this paper, we propose and discuss the usage of the false discovery rate in the unsupervised BMF setting. We prove two bounds on the p…
University of Bergen（卑尔根大学）；IIT Hyderabad（印度理工学院海得拉巴分校）；EdgeVerve Systems Limited（EdgeVerve系统有限公司）；Amazon（亚马逊）
Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms…
KTH Royal Institute of Technology（瑞典皇家理工学院）；University of Vienna（维也纳大学）；Brown University（布朗大学）；University of Eastern Finland（东芬兰大学）
We study the clustering of bipartite graphs and Boolean matrix factorization in data streams. We consider a streaming setting in which the vertices from the left side of the graph arrive one by one together with all of their incident edges. We provide an algorithm that, after one pass over the stream, recovers the set of clusters on the right side of the graph using sublinear space; to the best of our knowledge, this is the first algorithm with this property. We also show that after a second pass over the stream, the left clusters of the bipartite graph can be recovered and we show how to exte…
A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with the constraint over an auxiliary matrix whose Boolean structure is identical to the initial Boolean data. Then the solution of the nonnegative auxiliary optimization problem is thresholded to provide a solution for the BMF problem. We provide the proofs for the equivalencies of the two solution spaces under the existence of an exact solution. Moreover, the nonincreasing property of the algorithm is also proven. Experiment…
A Bayesian Boolean Matrix Factorization with Application to Copy Number Analysis in Cancer
Department of Data Science, Dana-Farber Cancer Institute（数据科学部，达纳-法伯癌症研究所）；Department of Biostatistics, Harvard T.H. Chan School of Public Health（生物统计学部，哈佛T.H. 潘克学校公共卫生学院）
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 Boo…
University of Helsinki（赫尔辛基大学）；University of Eastern Finland（东芬兰大学）；MPI-INF（马克斯·普朗克信息学研究所）
Boolean matrix factorization is a natural and a popular technique for summarizing binary matrices. In this paper, we study a problem of Boolean matrix factorization where we additionally require that the factor matrices have consecutive ones property (OBMF). A major application of this optimization problem comes from graph visualization: standard techniques for visualizing graphs are circular or linear layout, where nodes are ordered in circle or on a line. A common problem with visualizing graphs is clutter due to too many edges. The standard approach to deal with this is to bundle edges toge…
Matrix Factorization for Inferring Associations and Missing Links
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）；Institute for Clarity in Documentation（文献清晰度研究所）；Inria Paris-Rocquencourt（法国国家信息与自动化研究所巴黎-罗康校区）；Rajiv Gandhi University（拉吉夫·甘地大学）；Tsinghua University（清华大学）；Palmer Research Laboratories（帕尔默研究实验室）
In this work, we study a variant of nonnegative matrix factorization where we wish to find a symmetric factorization of a given input matrix into a sparse, Boolean matrix. Formally speaking, given $\mathbf{M}\in\mathbb{Z}^{m\times m}$, we want to find $\mathbf{W}\in\{0,1\}^{m\times r}$ such that $\| \mathbf{M} - \mathbf{W}\mathbf{W}^\top \|_0$ is minimized among all $\mathbf{W}$ for which each row is $k$-sparse. This question turns out to be closely related to a number of questions like recovering a hypergraph from its line graph, as well as reconstruction attacks for private neural network tr…
Mining and exploring databases should provide users with knowledge and new insights. Tiles of data strive to unveil true underlying structure and distinguish valuable information from various kinds of noise. We propose a novel Boolean matrix factorization algorithm to solve the tiling problem, based on recent results from optimization theory. In contrast to existing work, the new algorithm minimizes the description length of the resulting factorization. This approach is well known for model selection and data compression, but not for finding suitable factorizations via numerical optimization. …
Boolean tensor has been broadly utilized in representing high dimensional logical data collected on spatial, temporal and/or other relational domains. Boolean Tensor Decomposition (BTD) factorizes a binary tensor into the Boolean sum of multiple rank-1 tensors, which is an NP-hard problem. Existing BTD methods have been limited by their high computational cost, in applications to large scale or higher order tensors. In this work, we presented a computationally efficient BTD algorithm, namely \textit{Geometric Expansion for all-order Tensor Factorization} (GETF), that sequentially identifies th…
Federated Binary Matrix Factorization using Proximal Optimization
KTH Royal Institute of Technology（瑞典皇家理工学院）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）；Institute for AI in Medicine, UK Essen（埃森大学医院人工智能医学研究所）；Ruhr University Bochum（波鸿鲁尔大学）；Monash University（莫纳什大学）
Inria Saclay（法国国家信息与自动化研究所萨克莱分所）；Univ. Paris Saclay（巴黎萨克雷大学）；IIIT - Delhi（德里英迪拉·甘地信息技术学院）
Co-administration of two or more drugs simultaneously can result in adverse drug reactions. Identifying drug-drug interactions (DDIs) is necessary, especially for drug development and for repurposing old drugs. DDI prediction can be viewed as a matrix completion task, for which matrix factorization (MF) appears as a suitable solution. This paper presents a novel Graph Regularized Probabilistic Matrix Factorization (GRPMF) method, which incorporates expert knowledge through a novel graph-based regularization strategy within an MF framework. An efficient and sounded optimization algorithm is pro…
Computational drug repositioning aims to discover new therapeutic diseases for marketed drugs and has the advantages of low cost, short development cycle, and high controllability compared to traditional drug development. The matrix factorization model has become the cornerstone technique for computational drug repositioning due to its ease of implementation and excellent scalability. However, the matrix factorization model uses the inner product to represent the association between drugs and diseases, which is lacking in expressive ability. Moreover, the degree of similarity of drugs or disea…
Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing…
What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Eindhoven University of Technology（埃因霍温理工大学）；Graz University of Technology（格拉茨工业大学）；University of Bari Aldo Moro（巴里大学 Aldo Moro）
Beihang University（北京航空航天大学）；Microsoft Research Asia（微软亚洲研究院）
Recent years have witnessed the world-wide emergence of mega-metropolises with incredibly huge populations. Understanding residents mobility patterns, or urban dynamics, thus becomes crucial for building modern smart cities. In this paper, we propose a Neighbor-Regularized and context-aware Non-negative Tensor Factorization model (NR-cNTF) to discover interpretable urban dynamics from urban heterogeneous data. Different from many existing studies concerned with prediction tasks via tensor completion, NR-cNTF focuses on gaining urban managerial insights from spatial, temporal, and spatio-tempor…
Real-world item recommenders commonly suffer from a persistent cold-start problem which is caused by dynamically changing users and items. In order to overcome the problem, several context-aware recommendation techniques have been recently proposed. In terms of both feasibility and performance, factorization machine (FM) is one of the most promising methods as generalization of the conventional matrix factorization techniques. However, since online algorithms are suitable for dynamic data, the static FMs are still inadequate. Thus, this paper proposes incremental FMs (iFMs), a general online f…
Rutgers, The State University of New Jersey（罗格斯新泽西州立大学）
In many signal processing and machine learning applications, datasets containing private information are held at different locations, requiring the development of distributed privacy-preserving algorithms. Tensor and matrix factorizations are key components of many processing pipelines. In the distributed setting, differentially private algorithms suffer because they introduce noise to guarantee privacy. This paper designs new and improved distributed and differentially private algorithms for two popular matrix and tensor factorization methods: principal component analysis (PCA) and orthogonal…
If learning methods are to scale to the massive sizes of modern datasets, it is essential for the field of machine learning to embrace parallel and distributed computing. Inspired by the recent development of matrix factorization methods with rich theory but poor computational complexity and by the relative ease of mapping matrices onto distributed architectures, we introduce a scalable divide-and-conquer framework for noisy matrix factorization. We present a thorough theoretical analysis of this framework in which we characterize the statistical errors introduced by the "divide" step and cont…
We propose a general algorithmic framework for constrained matrix and tensor factorization, which is widely used in signal processing and machine learning. The new framework is a hybrid between alternating optimization (AO) and the alternating direction method of multipliers (ADMM): each matrix factor is updated in turn, using ADMM, hence the name AO-ADMM. This combination can naturally accommodate a great variety of constraints on the factor matrices, and almost all possible loss measures for the fitting. Computation caching and warm start strategies are used to ensure that each update is eva…
Data-driven decision making is rapidly gaining popularity, fueled by the ever-increasing amounts of available data and encouraged by the development of models that can identify beyond linear input-output relationships. Simultaneously the need for interpretable prediction- and classification methods is increasing, as this improves both our trust in these models and the amount of information we can abstract from data. An important aspect of this interpretability is to obtain insight in the sensitivity-specificity trade-off constituted by multiple plausible input-output relationships. These are o…
