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 structure which corresponds to our model assumption, and on the other hand that our model assumption is justified in real-world application. Our method is parameter-free.
Miettinen @cite_12 transfers the objective of JSMF into Boolean algebra, solving [ X,Y a 1,2 2 | - ( ) | ] for binary matrices @math and @math , and normalizing constants @math . A variant of the BMF algorithm @cite_10 governs the minimization. A provisional determination of ranks based on the Minimum Description Length (MDL) principle is proposed, computing which of the candidate rank constellations yields the lowest description length. The description length captures model complexity and data fit, and is hence suitable for model order selection @cite_6 @cite_14 .
@cite_14 @cite_10 @cite_6 @cite_12
2915000964 60908300 1986170391 2151381597
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. In general, binary matrix factorization (BMF) refers to the problem of finding two binary matrices of low rank such that the difference between their matrix product and a given binary matrix is minimal. BMF has served as an important tool in dimension reduction for high-dimensional data sets with binary attributes and has been successfully employed in numerous applications. In the existing literature on BMF, the matrix product is not required to be binary. We call this unconstrained BMF (UBMF) and similarly constrained BMF (CBMF) if the matrix product is required to be binary. In this paper, we first introduce two specific variants of CBMF and discuss their relation to other dimensional reduction models such as UBMF. Then we propose alternating update procedures for CBMF. In every iteration of the proposed procedure, we solve a specific binary linear programming (BLP) problem to update the involved matrix argument. We explore the relationship between the BLP subproblem and clustering to develop an effective 2- approximation algorithm for CBMF when the underlying matrix has very low rank. The proposed algorithm can also provide a 2-approximation to rank-1 UBMF. We also develop a randomized algorithm for CBMF and estimate the approximation ratio of the solution obtained. Numerical experiments show that the proposed algorithm for UBMF finds better solutions in less CPU time than several other algorithms in the literature, and the solution obtained from CBMF is very close to that of UBMF. 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 show that the maximization of the sum degrees-of-freedom for the static flat-fading multiple-input multiple-output (MIMO) interference channel is equivalent to a rank constrained rank minimization problem (RCRM), when the signal spaces span all available dimensions. The rank minimization corresponds to maximizing interference alignment (IA) so that interference spans the lowest dimensional subspace possible. The rank constraints account for the useful signal spaces spanning all available spatial dimensions. That way, we reformulate all IA requirements to requirements involving ranks. Then, we present a convex relaxation of the RCRM problem inspired by recent results in compressed sensing and low-rank matrix completion theory that rely on approximating rank with the nuclear norm. We show that the convex envelope of the sum of ranks of the interference matrices is the normalized sum of their corresponding nuclear norms and introduce tractable constraints that are asymptotically equivalent to the rank constraints for the initial problem. We also show that our heuristic relaxation can be tuned for the multi-cell interference channel. Furthermore, we experimentally show that in many cases the proposed algorithm attains perfect interference alignment and in some cases outperforms previous approaches for finding precoding and zero-forcing matrices for interference alignment.