  The blind deconvolution problem aims to recover a rank-one matrix from a set
of rank-one linear measurements. Recently, Charisopulos et al. introduced a
nonconvex nonsmooth formulation that can be used, in combination with an
initialization procedure, to provably solve this problem under standard
statistical assumptions. In practice, however, initialization is unnecessary.
As we demonstrate numerically, a randomly initialized subgradient method
consistently solves the problem. In pursuit of a better understanding of this
phenomenon, we study the random landscape of this formulation. We characterize
in closed form the landscape of the population objective and describe the
approximate location of the stationary points of the sample objective. In
particular, we show that the set of spurious critical points lies close to a
codimension two subspace. In doing this, we develop tools for studying the
landscape of a broader family of singular value functions, these results may be
of independent interest.
