Alternating projections on manifolds

We prove that if two smooth manifolds intersect transversally, then the method of alternating projections converges locally at a linear rate. We bound the speed of convergence in terms of the angle between the manifolds, which in turn we relate to the modulus of metric regularity for the intersection problem, a natural measure of conditioning. We discuss a variety of problem classes where the projections are com-putationally tractable, and we illustrate the method numerically on a problem of finding a low-rank solution of a matrix equation. Key words: alternating projections, nonconvex, linear convergence, sub-space angle, metric regularity, low-rank approximation, spectral set AMS 2000 Subject Classification: 49M29, 65K10, 90C30