Convergence Analysis of Alternating Projection Method for Nonconvex Sets

TL;DR

This paper provides a new convergence analysis framework for the alternating projection method on nonconvex sets, establishing conditions under which the method converges to a critical point and deriving convergence rates.

Contribution

It introduces properties of semi-algebraic sets and leverages the Kurdyka-oshiewicz (KL) property to analyze convergence, recovering known rates and applying to structured tight frame design.

Findings

Convergence of alternating projection under three-point and local contraction properties.

Explicit convergence rate depending on the KL exponent.

Application to structured tight frame design with convergence guarantees.

Abstract

Alternating projection method has been used in a wide range of engineering applications since it is a gradient-free method (without requiring tuning the step size) and usually has fast speed of convergence. 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 close enough to each other. Then by exploiting the geometric properties of the objective function around its critical point, i.e. the Kurdyka-\L{ojasiewicz} (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 projection method 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 projection method onto closed convex sets. To illustrate the power of our new framework, we provide new convergence result for a class of concrete applications: alternating projection method for designing structured tight frames that are widely used in sparse representation, compressed sensing and communication.