Scaled Gradient Descent for Ill-Conditioned Low-Rank Matrix Recovery with Optimal Sampling Complexity

The low-rank matrix recovery problem seeks to reconstruct an unknown $n_1\times n_2$ rank-$r$ matrix from $m$ linear measurements, where $m\ll n_1{n}_2$. This problem has been extensively studied over the past few decades, leading to a variety of algorithms with solid theoretical guarantees. Among these, gradient descent based non-convex methods have become particularly popular due to their computational efficiency. However, these methods typically suffer from two key limitations: a sub-optimal sample complexity of $O((n_1+n_2)r^2)$ and an iteration complexity of $O(\kappa \log (1\mathrm{/}\epsilon ))$ to achieve $\epsilon$-accuracy, resulting in slow convergence when the target matrix is ill-conditioned. Here, $\kappa$ denotes the condition number of the unknown matrix. Recent studies show that a preconditioned variant of GD, known as scaled gradient descent (ScaledGD), can significantly reduce the iteration complexity to $O(\log (1\mathrm{/}\epsilon ))$. Nonetheless, its sample complexity remains sub-optimal at $O((n_1+n_2)r^2)$. In contrast, a delicate virtual sequence technique demonstrates that the standard GD in the positive semidefinite (PSD) setting achieves the optimal sample complexity $O((n_1+n_2)r)$, but converges more slowly with an iteration complexity $O({\kappa }^2\log (1\mathrm{/}\epsilon ))$. In this paper, through a more refined analysis, we show that ScaledGD achieves both the optimal sample complexity $O((n_1+n_2)r)$ and the improved iteration complexity $O(\log (1\mathrm{/}\epsilon ))$. Notably, our results extend beyond the PSD setting to general low-rank matrix recovery problem. Numerical experiments further validate that ScaledGD accelerates convergence for ill-conditioned matrices with the optimal sampling complexity.