Convergence Analysis of Alternating Nonconvex Projections

We consider the convergence properties for alternating projection algorithm (a.k.a alternating projections) which has been widely utilized to solve many practical problems in machine learning, signal and image processing, communication and statistics since it is a gradient-free method (without requiring tuning the step size) and it usually converges fast for solving practical problems. Although alternating projections works surprisingly well in practice, including for nonconvex nonsmooth sets in some applications, the theoretical understanding of the alternating projections is still an active research area, especially the convergence behavior. In this paper, we formalize two properties of proper, lower semi-continuous and semi-algebraic sets: the three point property for all possible iterates and the local contraction property that serves as the non-expensiveness property of the projector, but only for the iterates that are closed enough to each other. Then by exploiting the geometric properties of the objective function around its critical point, i.e. the KurdykaLojasiewicz (KL) property, we establish a new convergence analysis framework to show that if one set satisfies the three point property and the other one obeys the local contraction property, the iterates generated by alternating projections is a convergent sequence and converges to a critical point. We complete this study by providing convergence rate which depends on the explicit expression of the KL exponent. As a byproduct, we use our new analysis framework to recover the linear convergence rate of alternating projections onto closed convex sets. To illustrate the power of our new framework, we provide new convergence result for a class of concrete applications: alternating projections for designing structured tight frames that are widely used in sparse representation, compressed sensing and communication. We believe that our new analysis framework can be applied to guarantee the convergence of alternating projections when utilized for many other nonconvex and nonsmooth sets.