  Matrix completion aims to reconstruct a data matrix based on observations of
a small number of its entries. Usually in matrix completion a single matrix is
considered, which can be, for example, a rating matrix in recommendation
system. However, in practical situations, data is often obtained from multiple
sources which results in a collection of matrices rather than a single one. In
this work, we consider the problem of collective matrix completion with
multiple and heterogeneous matrices, which can be count, binary, continuous,
etc. We first investigate the setting where, for each source, the matrix
entries are sampled from an exponential family distribution. Then, we relax the
assumption of exponential family distribution for the noise and we investigate
the distribution-free case. In this setting, we do not assume any specific
model for the observations. The estimation procedures are based on minimizing
the sum of a goodness-of-fit term and the nuclear norm penalization of the
whole collective matrix. We prove that the proposed estimators achieve fast
rates of convergence under the two considered settings and we corroborate our
results with numerical experiments.
