  We consider convex optimization problems which are widely used as convex
relaxations for low-rank matrix recovery problems. In particular, in several
important problems, such as phase retrieval and robust PCA, the underlying
assumption in many cases is that the optimal solution is rank-one. In this
paper we consider a simple and natural sufficient condition on the objective so
that the optimal solution to these relaxations is indeed unique and rank-one.
Mainly, we show that under this condition, the standard Frank-Wolfe method with
line-search (i.e., without any tuning of parameters whatsoever), which only
requires a single rank-one SVD computation per iteration, finds an
$\epsilon$-approximated solution in only $O(\log{1/\epsilon})$ iterations (as
opposed to the previous best known bound of $O(1/\epsilon)$), despite the fact
that the objective is not strongly convex. We consider several variants of the
basic method with improved complexities, as well as an extension motivated by
robust PCA, and finally, an extension to nonsmooth problems.
