  The problem of completing high-dimensional matrices from a limited set of
observations arises in many big data applications, especially, recommender
systems. Existing matrix completion models generally follow either a memory- or
a model-based approach, whereas, geometric matrix completion models combine the
best from both approaches. Existing deep-learning-based geometric models yield
good performance, but, in order to operate, they require a fixed structure
graph capturing the relationships among the users and items. This graph is
typically constructed by evaluating a pre-defined similarity metric on the
available observations or by using side information, e.g., user profiles. In
contrast, Markov-random-fields-based models do not require a fixed structure
graph but rely on handcrafted features to make predictions. When no side
information is available and the number of available observations becomes very
low, existing solutions are pushed to their limits. In this paper, we propose a
geometric matrix completion approach that addresses these challenges. We
consider matrix completion as a structured prediction problem in a conditional
random field (CRF), which is characterized by a maximum a posterior (MAP)
inference, and we propose a deep model that predicts the missing entries by
solving the MAP inference problem. The proposed model simultaneously learns the
similarities among matrix entries, computes the CRF potentials, and solves the
inference problem. Its training is performed in an end-to-end manner, with a
method to supervise the learning of entry similarities. Comprehensive
experiments demonstrate the superior performance of the proposed model compared
to various state-of-the-art models on popular benchmark datasets and underline
its superior capacity to deal with highly incomplete matrices.
