Accelerating Gradient Descent for Over-Parameterized Asymmetric Low-Rank Matrix Sensing via Preconditioning

We present an accelerated method for the asymmetric low-rank matrix sensing problem in the over-parameterized setup, named preconditioned gradient descent. We analyze the local convergence rate of the proposed algorithm starting from spectral initialization. Our algorithm is shown to have linear convergence rate independent of condition number even when ill-conditioning and over-parameterization both exist in the asymmetric matrix sensing problem. Numerical results verify the theoretical findings and demonstrate the performance of the proposed algorithm.