Local Linear Convergence for Inexact Alternating Projections on Nonconvex Sets

Abstract

Given two arbitrary closed sets in Euclidean space, to guarantee that the method of alternating projections converges locally at linear rate to a point in the intersection, a simple transversality assumption suffices. Exact projection onto nonconvex sets is typically intractable, but we show that computationally cheap inexact projections may suffice instead. In particular, if one set is defined by sufficiently regular smooth constraints, then projecting onto the approximation obtained by linearizing those constraints around the current iterate suffices. On the other hand, if one set is a smooth manifold represented through local coordinates, then the approximate projection resulting from linearizing the coordinate system around the preceding iterate on the manifold also suffices.