  Non-convex gradient descent is a common approach for estimating a low-rank
$n\times n$ ground truth matrix from noisy measurements, because it has
per-iteration costs as low as $O(n)$ time, and is in theory capable of
converging to a minimax optimal estimate. However, the practitioner is often
constrained to just tens to hundreds of iterations, and the slow and/or
inconsistent convergence of non-convex gradient descent can prevent a
high-quality estimate from being obtained. Recently, the technique of
preconditioning was shown to be highly effective at accelerating the local
convergence of non-convex gradient descent when the measurements are noiseless.
In this paper, we describe how preconditioning should be done for noisy
measurements to accelerate local convergence to minimax optimality. For the
symmetric matrix sensing problem, our proposed preconditioned method is
guaranteed to locally converge to minimax error at a linear rate that is immune
to ill-conditioning and/or over-parameterization. Using our proposed
preconditioned method, we perform a 60 megapixel medical image denoising task,
and observe significantly reduced noise levels compared to previous approaches.
