Alternating least squares as moving subspace correction
For researchers in numerical linear algebra and optimization, this offers a conceptually simple derivation of known convergence rates, but the abstract conditions remain unverified for most cases, making it an incremental contribution.
The paper provides a new interpretation of alternating optimization methods for low-rank matrices and tensors as sequential optimization on moving subspaces, and derives the asymptotic convergence rate of the two-sided block power method (equivalent to alternating least squares) for computing dominant singular subspaces.
In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimization on moving subspaces yields insightful reformulations of some known convergence conditions that focus on the interplay between the contractivity of classical multiplicative Schwarz methods with overlapping subspaces and the curvature of low-rank matrix and tensor manifolds. While the verification of the abstract conditions in concrete scenarios remains open in most cases, we are able to provide an alternative and conceptually simple derivation of the asymptotic convergence rate of the two-sided block power method of numerical algebra for computing the dominant singular subspaces of a rectangular matrix. This method is equivalent to an alternating least squares method applied to a distance function. The theoretical results are illustrated and validated by numerical experiments.