Edward Meeds, Zoubin Ghahramani, Radford Neal, Sam Roweis
We introduce binary matrix factorization, a novel model for unsupervised ma- trix decomposition. The decomposition is learned by ﬁtting a non-parametric Bayesian probabilistic model with binary latent variables to a matrix of dyadic data. Unlike bi-clustering models, which assign each row or column to a single cluster based on a categorical hidden feature, our binary feature model reﬂects the prior belief that items and attributes can be associated with more than one latent cluster at a time. We provide simple learning and inference rules for this new model and show how to extend it to an inﬁnite model in which the number of features is not a priori ﬁxed but is allowed to grow with the size of the data.
1 Distributed representations for dyadic data
One of the major goals of probabilistic unsupervised learning is to discover underlying or hidden structure in a dataset by using latent variables to describe a complex data generation process. In this paper we focus on dyadic data: our domains have two ﬁnite sets of objects/entities and observa- tions are made on dyads (pairs with one element from each set). Examples include sparse matrices of movie-viewer ratings, word-document counts or product-customer purchases. A simple way to capture structure in this kind of data is to do “bi-clustering” (possibly using mixture models) by grouping the rows and (independently or simultaneously) the columns[6, 13, 9]. The modelling as-
sumption in such a case is that movies come in types and viewers in types and that knowing componential structure: each item (row) has associated with it an unobserved vector of binary features; similarly each attribute (column) has a hidden vector of binary features. Knowing the matrixX into (a distribution deﬁned by) the productUWV> , whereU andV are binary feature matrices, andW is a real-valued weight matrix. Below, we develop this binary matrix factorization
the type of movie and type of viewer is sufﬁcient to predict the response. Clustering or mixture models are quite restrictive – their major disadvantage is that they do not admit a componential or distributed representation because items cannot simultaneously belong to several classes. (A movie, for example, might be explained as coming from a cluster of “dramas” or “comedies”; a viewer as a “single male” or as a “young mother”.) We might instead prefer a model (e.g. [10, 5]) in which objects can be assigned to multiple latent clusters: a movie might be a drama and have won an Os- car and have subtitles; a viewer might be single and female and a university graduate. Inference in such models falls under the broad area of factorial learning (e.g. [7, 1, 3, 12]), in which multiple interacting latent causes explain each observed datum.
features of the item and the features of the attribute are sufﬁcient to generate (before noise) the response at that location in the matrix. In effect, we are factorizing a real-valued data (response)
In this paper, we assume that both data items (rows) and attributes (columns) have this kind of