  We present theoretical results on the convergence of \emph{non-convex}
accelerated gradient descent in matrix factorization models with $\ell_2$-norm
loss. The purpose of this work is to study the effects of acceleration in
non-convex settings, where provable convergence with acceleration should not be
considered a \emph{de facto} property. The technique is applied to matrix
sensing problems, for the estimation of a rank $r$ optimal solution $X^\star
\in \mathbb{R}^{n \times n}$. Our contributions can be summarized as follows.
$i)$ We show that acceleration in factored gradient descent converges at a
linear rate; this fact is novel for non-convex matrix factorization settings,
under common assumptions. $ii)$ Our proof technique requires the acceleration
parameter to be carefully selected, based on the properties of the problem,
such as the condition number of $X^\star$ and the condition number of objective
function. $iii)$ Currently, our proof leads to the same dependence on the
condition number(s) in the contraction parameter, similar to recent results on
non-accelerated algorithms. $iv)$ Acceleration is observed in practice, both in
synthetic examples and in two real applications: neuronal multi-unit activities
recovery from single electrode recordings, and quantum state tomography on
quantum computing simulators.
