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Alternating Projections on Manifolds

School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853
INRIA Rhone-Alpes, Montbonnot, 38334 St. Ismier Cedex, France
aslewis{at}orie.cornell.edu
jerome.malick{at}inria.fr

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 computationally 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; subspace angle; metric regularity; low-rank approximation; spectral set
History: Received: July 27, 2006; revision received: June 13, 2007;