  The problem of finding the missing values of a matrix given a few of its
entries, called matrix completion, has gathered a lot of attention in the
recent years. Although the problem under the standard low rank assumption is
NP-hard, Cand\`es and Recht showed that it can be exactly relaxed if the number
of observed entries is sufficiently large. In this work, we introduce a novel
matrix completion model that makes use of proximity information about rows and
columns by assuming they form communities. This assumption makes sense in
several real-world problems like in recommender systems, where there are
communities of people sharing preferences, while products form clusters that
receive similar ratings. Our main goal is thus to find a low-rank solution that
is structured by the proximities of rows and columns encoded by graphs. We
borrow ideas from manifold learning to constrain our solution to be smooth on
these graphs, in order to implicitly force row and column proximities. Our
matrix recovery model is formulated as a convex non-smooth optimization
problem, for which a well-posed iterative scheme is provided. We study and
evaluate the proposed matrix completion on synthetic and real data, showing
that the proposed structured low-rank recovery model outperforms the standard
matrix completion model in many situations.
