One-Class Matrix Completion with Low-Density Factorizations

Consider a typical recommendation problem. A company has historical records of products sold to a large customer base. These records may be compactly represented as a sparse customer-times-product ``who-bought-what" binary matrix. Given this matrix, the goal is to build a model that provides recommendations for which products should be sold next to the existing customer base. Such problems may naturally be formulated as collaborative filtering tasks. However, this is a {\it one-class} setting, that is, the only known entries in the matrix are one-valued. If a customer has not bought a product yet, it does not imply that the customer has a low propensity to {\it potentially} be interested in that product. In the absence of entries explicitly labeled as negative examples, one may resort to considering unobserved customer-product pairs as either missing data or as surrogate negative instances. In this paper, we propose an approach to explicitly deal with this kind of ambiguity by instead treating the unobserved entries as optimization variables. These variables are optimized in conjunction with learning a weighted, low-rank non-negative matrix factorization (NMF) of the customer-product matrix, similar to how Transductive SVMs implement the low-density separation principle for semi-supervised learning. Experimental results show that our approach gives significantly better recommendations in comparison to various competing alternatives on one-class collaborative filtering tasks.