  In this note, we focus on smooth nonconvex optimization problems that obey:
(1) all local minimizers are also global; and (2) around any saddle point or
local maximizer, the objective has a negative directional curvature. Concrete
applications such as dictionary learning, generalized phase retrieval, and
orthogonal tensor decomposition are known to induce such structures. We
describe a second-order trust-region algorithm that provably converges to a
global minimizer efficiently, without special initializations. Finally we
highlight alternatives, and open problems in this direction.
