  We consider solving the low rank matrix sensing problem with Factorized
Gradient Descend (FGD) method when the true rank is unknown and over-specified,
which we refer to as over-parameterized matrix sensing. If the ground truth
signal $\mathbf{X}^* \in \mathbb{R}^{d*d}$ is of rank $r$, but we try to
recover it using $\mathbf{F} \mathbf{F}^\top$ where $\mathbf{F} \in
\mathbb{R}^{d*k}$ and $k>r$, the existing statistical analysis falls short, due
to a flat local curvature of the loss function around the global maxima. By
decomposing the factorized matrix $\mathbf{F}$ into separate column spaces to
capture the effect of extra ranks, we show that $\|\mathbf{F}_t \mathbf{F}_t -
\mathbf{X}^*\|_{F}^2$ converges to a statistical error of $\tilde{\mathcal{O}}
({k d \sigma^2/n})$ after
$\tilde{\mathcal{O}}(\frac{\sigma_{r}}{\sigma}\sqrt{\frac{n}{d}})$ number of
iterations where $\mathbf{F}_t$ is the output of FGD after $t$ iterations,
$\sigma^2$ is the variance of the observation noise, $\sigma_{r}$ is the $r$-th
largest eigenvalue of $\mathbf{X}^*$, and $n$ is the number of sample. Our
results, therefore, offer a comprehensive picture of the statistical and
computational complexity of FGD for the over-parameterized matrix sensing
problem.
