  The blind deconvolution problem seeks to recover a pair of vectors from a set
of rank one bilinear measurements. We consider a natural nonsmooth formulation
of the problem and show that under standard statistical assumptions, its moduli
of weak convexity, sharpness, and Lipschitz continuity are all dimension
independent. This phenomenon persists even when up to half of the measurements
are corrupted by noise. Consequently, standard algorithms, such as the
subgradient and prox-linear methods, converge at a rapid dimension-independent
rate when initialized within constant relative error of the solution. We then
complete the paper with a new initialization strategy, complementing the local
search algorithms. The initialization procedure is both provably efficient and
robust to outlying measurements. Numerical experiments, on both simulated and
real data, illustrate the developed theory and methods.
