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.
The BMF problem is closely connected, if not identical, to many other computational problems including dense bipartite subgraph extraction ( @cite_8 , @cite_7 ), the tiling problem ( @cite_6 ), and the binary independent component analysis ( @cite_9 ). Various heuristics has been adopted to develop efficient approximates, including the discrete basis problem ( @cite_16 ), linear programming ( @cite_5 ), formal concept analysis ( @cite_2 , @cite_1 ) and minimum description length ( @cite_17 , @cite_4 ). However, most algorithms cannot handle both noise and missing values. For example, @cite_7 presented a simple two-step algorithm that can identify tiny clusters on the right side even under destructive noise level. But it assumed the cluster size on the left are large while those on the right are small. Moreover, it cannot handle missing values.
@cite_4 @cite_7 @cite_8 @cite_9 @cite_1 @cite_6 @cite_2 @cite_5 @cite_16 @cite_17
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Matrix factorizations---where a given data matrix is approximated by a product of two or more factor matrices---are powerful data mining tools. Among other tasks, matrix factorizations are often used to separate global structure from noise. This, however, requires solving the model order selection problem' of determining where fine-grained structure stops, and noise starts, i.e., what is the proper size of the factor matrices. Boolean matrix factorization (BMF)---where data, factors, and matrix product are Boolean---has received increased attention from the data mining community in recent years. The technique has desirable properties, such as high interpretability and natural sparsity. But so far no method for selecting the correct model order for BMF has been available. In this paper we propose to use the Minimum Description Length (MDL) principle for this task. Besides solving the problem, this well-founded approach has numerous benefits, e.g., it is automatic, does not require a likelihood function, is fast, and, as experiments show, is highly accurate. We formulate the description length function for BMF in general---making it applicable for any BMF algorithm. We extend an existing algorithm for BMF to use MDL to identify the best Boolean matrix factorization, analyze the complexity of the problem, and perform an extensive experimental evaluation to study its behavior. We study the problem of finding planted clusters in bipartite graphs. We present a simple two-step algorithm which provably finds even tiny clusters of size O(n^e), where n is the number of vertices in the graph and e > 0. Previous algorithms were only able to identify clusters of size Ω( sqrt(n) ). We evaluated the algorithm on synthetic and on real-world data; the experiments show that the algorithm can find extremely small clusters even in presence of high destructive noise. We present a framework for biclustering and clustering where the observations are general labels. Our approach is based on the maximum likelihood estimator and its convex relaxation, and generalizes recent works in graph clustering to the biclustering setting. In addition to standard biclustering setting where one seeks to discover clustering structure simultaneously in two domain sets, we show that the same algorithm can be as effective when clustering structure only occurs in one domain. This allows for an alternative approach to clustering that is more natural in some scenarios. We present theoretical results that provide sufficient conditions for the recovery of the true underlying clusters under a generalized stochastic block model. These are further validated by our empirical results on both synthetic and real data. Independent component analysis (ICA) is a computational method for separating a multivariate signal into subcomponents assuming the mutual statistical independence of the non-Gaussian source signals. The classical independent components analysis (ICA) framework usually assumes linear combinations of independent sources over the field of real-valued numbers R. In this paper, we investigate binary ICA for or mixtures (bICA), which can find applications in many domains including medical diagnosis, multi-cluster assignment, Internet tomography and network resource management. We prove that bICA is uniquely identifiable under the disjunctive generation model, and propose a deterministic iterative algorithm to determine the distribution of the latent random variables and the mixing matrix. The inverse problem to infer the values of latent variables is also considered for noisy measurements. We conduct an extensive simulation study to verify the effectiveness of the propose algorithm and present examples of real-world applications where bICA can be applied. Abstract We present new results on Boolean matrix factorization and a new algorithm based on these results. The results emphasize the significance of factorizations that provide from-below approximations of the input matrix. While the previously proposed algorithms do not consider the possibly different significance of different matrix entries, our results help measure such significance and suggest where to focus when computing factors. An experimental evaluation of the new algorithm on both synthetic and real data demonstrates its good performance in terms of good coverage by the first few factors as well as a small number of factors needed for an almost exact decomposition and indicates that the algorithm outperforms the available ones in these terms. We also propose future research topics. We propose a new approach for Collaborative Filtering which is based on Boolean Matrix Factorisation (BMF) and Formal Concept Analysis. In a series of experiments on real data (Movielens dataset) we compare the approach with the SVD- and NMF-based algorithms in terms of Mean Average Error (MAE). One of the experimental consequences is that it is enough to have a binary-scaled rating data to obtain almost the same quality in terms of MAE by BMF than for the SVD-based algorithm in case of non-scaled data. A decomposition of a binary matrix into two matrices gives a set of basis vectors and their appropriate combination to form the original matrix. Such decomposition solutions are useful in a number of application domains including text mining, role engineering as well as knowledge discovery. While a binary matrix can be decomposed in several ways, however, certain decompositions better characterize the semantics associated with the original matrix in a succinct but comprehensive way. Indeed, one can find different decompositions optimizing different criteria matching various semantics. In this paper, we first present a number of variants to the optimal Boolean matrix decomposition problem that have pragmatic implications. We then present a unified framework for modeling the optimal binary matrix decomposition and its variants using binary integer programming. Such modeling allows us to directly adopt the huge body of heuristic solutions and tools developed for binary integer programming. Although the proposed solutions are applicable to any domain of interest, for providing more meaningful discussions and results, in this paper, we present the binary matrix decomposition problem in a role engineering context, whose goal is to discover an optimal and correct set of roles from existing permissions, referred to as the role mining problem (RMP). This problem has gained significant interest in recent years as role based access control has become a popular means of enforcing security in databases. We consider several variants of the above basic RMP, including the min-noise RMP, delta-approximate RMP and edge-RMP. Solutions to each of them aid security administrators in specific scenarios. We then model these variants as Boolean matrix decomposition and present efficient heuristics to solve them. Matrix decomposition methods represent a data matrix as a product of two factor matrices: one containing basis vectors that represent meaningful concepts in the data, and another describing how the observed data can be expressed as combinations of the basis vectors. Decomposition methods have been studied extensively, but many methods return real-valued matrices. Interpreting real-valued factor matrices is hard if the original data is Boolean. In this paper, we describe a matrix decomposition formulation for Boolean data, the Discrete Basis Problem. The problem seeks for a Boolean decomposition of a binary matrix, thus allowing the user to easily interpret the basis vectors. We also describe a variation of the problem, the Discrete Basis Partitioning Problem. We show that both problems are NP-hard. For the Discrete Basis Problem, we give a simple greedy algorithm for solving it; for the Discrete Basis Partitioning Problem we show how it can be solved using existing methods. We present experimental results for the greedy algorithm and compare it against other, well known methods. Our algorithm gives intuitive basis vectors, but its reconstruction error is usually larger than with the real-valued methods. We discuss about the reasons for this behavior. 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 outperforms---from different standpoints---existing state-of-the-art BMF algorithms.