On the Computational and Statistical Complexity of Over-Parameterized Matrix Sensing

Abstract

We consider solving the low-rank matrix sensing problem with the Factorized Gradient Descent (FGD) method when the specified rank is larger than the true rank. We refer to this as over-parameterized matrix sensing. If the ground truth signal $\mathbf{X}^* \in \mathbb{R}^{d \times d}$ is of rank $r$, but we try to recover it using $\mathbf{F} \mathbf{F}^\top$ where $\mathbf{F} \in \mathbb{R}^{d \times k}$ and $k>r$, the existing statistical analysis either no longer holds or produces a vacuous statistical error upper bound (infinity) due to the 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 impact of using $k > r$, we show that $\left\| {\mathbf{F}_t \mathbf{F}_t - \mathbf{X}^*} \right\|_F^2$ converges sub-linearly to a statistical error of $\tilde{\mathcal{O}} (k d \sigma^2/n)$ after $\tilde{\mathcal{O}}(\frac{\sigma_{r}}{\sigma}\sqrt{\frac{n}{d}})$ 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 samples. With a precise characterization of the convergence behavior and the statistical error, our results, therefore, offer a comprehensive picture of the statistical and computational complexity if we solve the over-parameterized matrix sensing problem with FGD.