  We consider the matrix completion problem of recovering a structured matrix
from noisy and partial measurements. Recent works have proposed tractable
estimators with strong statistical guarantees for the case where the underlying
matrix is low--rank, and the measurements consist of a subset, either of the
exact individual entries, or of the entries perturbed by additive Gaussian
noise, which is thus implicitly suited for thin--tailed continuous data.
Arguably, common applications of matrix completion require estimators for (a)
heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b)
for heterogeneous noise models (beyond Gaussian), which capture varied
uncertainty in the measurements, and (c) heterogeneous structural constraints
beyond low--rank, such as block--sparsity, or a superposition structure of
low--rank plus elementwise sparseness, among others. In this paper, we provide
a vastly unified framework for generalized matrix completion by considering a
matrix completion setting wherein the matrix entries are sampled from any
member of the rich family of exponential family distributions; and impose
general structural constraints on the underlying matrix, as captured by a
general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized
$M$--estimator for the generalized framework, and provide a unified and novel
statistical analysis for this general class of estimators. We finally
corroborate our theoretical results on simulated datasets.
