Devising a complete and correct set of roles has been recognized as one of the most important and challenging tasks in implementing role based access control. A key problem related to this is the notion of goodness/interestingness – when is a role good/interesting? In this paper, we define the role mining problem (RMP) as the problem of discovering an optimal set of roles from existing user permissions. The main contribution of this paper is to formally define RMP, and analyze its theoretical bounds. In addition to the above basic RMP, we introduce two different variations of the RMP, called the δ-approx RMP and the Minimal Noise RMP that have pragmatic implications. We reduce the known “set basis problem ” to RMP to show that RMP is an NP-complete problem. An important contribution of this paper is also to show the relation of the role mining problem to several problems already identified in the data mining and data analysis literature. By showing that the RMP is in essence reducible to these known problems, we can directly borrow the existing implementation solutions and guide further research in this direction.

equally contributing corresponding author Conventional clustering methods typically assume that each data item belongs to a single cluster. This assumption does not hold in general. In order to overcome this limitation, we propose a generative method for clustering vectorial data, where each object can be assigned to multiple clusters. Using a deterministic annealing scheme, our method decomposes the observed data into the contributions of individual clusters and infers their parameters. Experiments on synthetic Boolean data show that our method achieves higher accuracy in the source parameter estimation and superior cluster stability compared to stateof-the-art approaches. We also apply our method to an important problem in computer security known as role mining. Experiments on real-world access control data show performance gains in generalization to new employees against other multi-assignment methods. In challenging situations with high noise levels, our approach maintains its good performance, while alternative state-of-the-art techniques lack robustness. 1.

We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of finding the maximum a posteriori (MAP) assignment of topics to words, where the document’s topic distribution is integrated out. We show that, when the effective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, finding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of finding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we briefly discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question. 1

Many 0/1 datasets have a very large number of variables; however, they are sparse and the dependency structure of the variables is simpler than the number of variables would suggest. Defining the effective dimensionality of such a dataset is a nontrivial problem. We consider the problem of defining a robust measure of dimension for 0/1 datasets, and show that the basic idea of fractal dimension can be adapted for binary data. However, as such the fractal dimension is difficult to interpret. Hence we introduce the concept of normalized fractal dimension. For a dataset D, its normalized fractal dimension counts the number of independent columns needed to achieve the unnormalized fractal dimension of D. The normalized fractal dimension measures the degree of dependency structure of the data. We study the properties of the normalized fractal dimension and discuss its computation. We give empirical results on the normalized fractal dimension, comparing it against PCA. 1.

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.

An interesting problem in Nonnegative Matrix Factorization (NMF) is to factorize the matrix X which is of some specific class, for example, binary matrix. In this paper, we extend the standard NMF to Binary Matrix Factorization (BMF for short): given a binary matrix X, we want to factorize X into two binary matrices W,H (thus conserving the most important integer property of the objective matrix X) satisfying X ≈ WH. Two algorithms are studied and compared. These methods rely on a fundamental boundedness property of NMF which we propose and prove. This new property also provides a natural normalization scheme that eliminates the bias of factor matrices. Experiments on both synthetic and real world datasets are conducted to show the competency and effectiveness of BMF. 1.

Preprint. To appear in Statistics and Computing. The final publication is available at www.springerlink.com. This paper proposes a hierarchical probabilistic model for ordinal matrix factorization. Unlike previous approaches, we model the ordinal nature of the data and take a principled approach to incorporating priors for the hidden variables. Two algorithms are presented for inference, one based on Gibbs sampling and one based on variational Bayes. Importantly, these algorithms may be implemented in the factorization of very large matrices with missing entries. The model is evaluated on a collaborative filtering task, where users have rated a collection of movies and the system is asked to predict their ratings for other movies. The Netflix data set is used for evaluation, which consists of around 100 million ratings. Using root mean-squared error (RMSE) as an evaluation metric, results show that the suggested model outperforms alternative factorization techniques. Results also show how Gibbs sampling outperforms variational Bayes on this task, despite the large number of ratings and model parameters. Matlab implementations of the proposed algorithms are available from cogsys.imm.dtu.dk/ordinalmatrixfactorization.

The discovery of patterns in binary dataset has many applications, e.g. in electronic commerce, TCP/IP networking, Web usage logging, etc. Still, this is a very challenging task in many respects: overlapping vs. non overlapping patterns, presence of noise, extraction of the most important patterns only. In this paper we formalize the problem of discovering the Top-K patterns from binary datasets in presence of noise, as the minimization of a novel cost function. According to the Minimum Description Length principle, the proposed cost function favors succinct pattern sets that may approximately describe the input data. We propose a greedy algorithm for the discovery of Patterns in Noisy Datasets, named PaNDa, and show that it outperforms related techniques on both synthetic and realworld data. 1

Abstract—Boolean matrix factorization is a method to decompose a binary matrix into two binary factor matrices. Akin to other matrix factorizations, the factor matrices can be used for various data analysis tasks. Many (if not most) real-world data sets are dynamic, though, meaning that new information is recorded over time. Incorporating this new information into the factorization can require a re-computation of the factorization – something we cannot do if we want to keep our factorization up-to-date after each update. This paper proposes a method to dynamically update the Boolean matrix factorization when new data is added to the data base. This method is extended with a mechanism to improve the factorization with a trade-off in speed of computation. The method is tested with a number of real-world and synthetic data sets including studying its efficiency against off-line methods. The results show that with good initialization the proposed online and dynamic methods can beat the stateof-the-art offline Boolean matrix factorization algorithms. Keywords-Boolean matrix factorization; On-line algorithms; Dynamic algorithms I.

Abstract — 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, δ-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. I.