  We formulate the problem of matrix completion with and without side
information as a non-convex optimization problem. We design fastImpute based on
non-convex gradient descent and show it converges to a global minimum that is
guaranteed to recover closely the underlying matrix while it scales to matrices
of sizes beyond $10^5 \times 10^5$. We report experiments on both synthetic and
real-world datasets that show fastImpute is competitive in both the accuracy of
the matrix recovered and the time needed across all cases. Furthermore, when a
high number of entries are missing, fastImpute is over $75\%$ lower in MAPE and
$15$ times faster than current state-of-the-art matrix completion methods in
both the case with side information and without.

Machine Learning;Optimization and Control;Methodology;Machine Learning