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We present theoretical results on the convergence of \emph{non-convex}accelerated gradient descent in matrix factorization models with $\ell_2$-normloss. The purpose of this work is to study the effects of acceleration innon-convex settings, where provable convergence with acceleration should not beconsidered a \emph{de facto} property. The technique is applied to matrixsensing 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 alinear rate; this fact is novel for non-convex matrix factorization settings,under common assumptions. $ii)$ Our proof technique requires the accelerationparameter to be carefully selected, based on the properties of the problem,such as the condition number of $X^\star$ and the condition number of objectivefunction. $iii)$ Currently, our proof leads to the same dependence on thecondition number(s) in the contraction parameter, similar to recent results onnon-accelerated algorithms. $iv)$ Acceleration is observed in practice, both insynthetic examples and in two real applications: neuronal multi-unit activitiesrecovery from single electrode recordings, and quantum state tomography onquantum computing simulators.

Provably convergent acceleration in factored gradient descent with applications in matrix sensing