Matrix factorizations—where a given data matrix is approximated by a prod- uct 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. However, 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 discuss how to construct an appropriate encoding, starting from a simple and intuitive approach, we arrive at a highly efficient data-to-model based encoding for BMF. 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.