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 extend our algorithm to solve the Boolean matrix factorization problem (by exploiting the correspondence of Boolean matrices and bipartite graphs). We evaluate an implementation of the algorithm on synthetic data and on real-world data. On real-world datasets the algorithm is orders of magnitudes faster than a static baseline algorithm while providing quality results within a factor 2 of the baseline algorithm. Our algorithm scales linearly in the number of edges in the graph. Finally, we analyze the algorithm theoretically and provide sufficient conditions under which the algorithm recovers a set of planted clusters under a standard random graph model.
The text-critical practice of grouping witnesses into families or texttypes often faces two obstacles: Contamination in the manuscript tradition, and co-dependence in identifying characteristic readings and manuscripts. We introduce non-negative matrix factorization (NMF) as a simple, unsupervised, and efficient way to cluster large numbers of manuscripts and readings simultaneously while summarizing contamination using an easy-to-interpret mixture model. We apply this method to an extensive collation of the New Testament epistle of Jude and show that the resulting clusters correspond to human…
University of Eastern Finland（东芬兰大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；University of Vienna（维也纳大学）
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 th…
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…
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 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…
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…
In this work we introduce a comprehensive algorithmic pipeline for multiple parametric model estimation. The proposed approach analyzes the information produced by a random sampling algorithm (e.g., RANSAC) from a machine learning/optimization perspective, using a \textit{parameterless} biclustering algorithm based on L1 nonnegative matrix factorization (L1-NMF). The proposed framework exploits consistent patterns that naturally arise during the RANSAC execution, while explicitly avoiding spurious inconsistencies. Contrarily to the main trends in the literature, the proposed technique does not…
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…
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…
Silesian University of Technology（西里西亚工业大学）；Pablo de Olavide University（巴勃罗·德·奥拉维德大学）；Universidad Nacional de San Agustín（圣奥古斯丁国立大学）
Biclustering is a powerful approach to search for patterns in data, as it can be driven by a function that measures the quality of diverse types of patterns of interest. However, due to its computational complexity, the exploration of the search space is usually guided by an algorithmic strategy, sometimes introducing random factors that simplify the computational cost (e.g. greedy search or evolutionary computation). Shifting patterns are specially interesting as they account constant fluctuations in data, i.e. they capture situations in which all the values in the pattern move up or down for…
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）
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…
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. …
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）
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…
Topic Modeling and Link-Prediction for Material Property Discovery
Theoretical Division, Los Alamos National Laboratory Los Alamos New Mexico USA；CSEE, University of Maryland, Baltimore County Baltimore Maryland USA；Modeling, Los Alamos National Laboratory Los Alamos New Mexico USA；Department of Material Science \& Engineering, University of Maryland College Park Maryland USA；Theoretical Division, Los Alamos National Laboratory；CSEE, University of Maryland, Baltimore County；Modeling, Los Alamos National Laboratory；Department of Material Science \& Engineering, University of Maryland
Matrix Completion in Group Testing: Bounds and Simulations
Faculty of Information Technology, University of Science, Ho Chi Minh city, Vietnam（信息科技学院，科学大学，胡志明市，越南）；Vietnam National University, Ho Chi Minh City, Vietnam（越南国家大学，胡志明市，越南）
University of Connecticut（康涅狄格大学）；University of Science and Technology of China（中国科学技术大学）
The sparse factorization of a large matrix is fundamental in modern statistical learning. In particular, the sparse singular value decomposition and its variants have been utilized in multivariate regression, factor analysis, biclustering, vector time series modeling, among others. The appeal of this factorization is owing to its power in discovering a highly-interpretable latent association network, either between samples and variables or between responses and predictors. However, many existing methods are either ad hoc without a general performance guarantee, or are computationally intensive…
