Linear convergence of Frank–Wolfe for rank-one matrix recovery without strong convexity

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