A Note on the Finite Convergence of Alternating Projections

TL;DR

This paper provides new conditions under which the alternating projections method converges in a finite number of steps for non-intersecting, potentially nonconvex sets, extending the concept of intrinsic transversality.

Contribution

It generalizes intrinsic transversality to non-intersecting sets and identifies minimal distance conditions for finite convergence in specific geometric cases.

Findings

Sufficient conditions for finite convergence of alternating projections.

Extension of intrinsic transversality to non-intersecting sets.

Minimal distance criteria for convergence in one iteration.

Abstract

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality, which until now has been restricted to sets with nonempty intersection. In the special case of a polyhedron and closed half space, our sufficient conditions define the minimum distance between the two sets that is required for alternating projections to converge in a single iteration.