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Geometry, Manifolds, and Nonconvex Optimization: How Geometry Can Help Optimization

Aimed at both casual spectators and active participants of optimization on manifolds alike, this introductory article presents a wide range of information I would have liked to have been told when I first entered the field. Several arguments are put forth: 1) it is not true that nonconvex implies difficult, 2) many optimization problems in signal processing are approached from the wrong perspective (once-off versus realtime optimization), and 3) the geometry of a manifold should not be used simply for the sake of it. This article also predicts that there is considerable potential for future work on optimization on compact manifolds; large classes of such problems are no harder than convex problems.