Journal article

Convergence Results For Projected Line-Search Methods On Varieties Of Low-Rank Matrices Via Lojasiewicz Inequality

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M-<= k of real mxn matrices of rank at most k. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold M-k of rank-k matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards projecting back to M-<= k. Considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of M-k. The pointwise convergence is obtained for real-analytic functions on the basis of a Lojasiewicz inequality for the projection of the antigradient to the tangent cone. If the derived limit point lies on the smooth part of M-<= k, i.e., in M-k, this boils down to more or less known results, but with the benefit that asymptotic convergence rate estimates (for specific step-sizes) can be obtained without an a priori curvature bound, simply from the fact that the limit lies on a smooth manifold. At the same time, one can give a convincing justification for assuming critical points to lie in M-k: if X is a critical point of f on M-<= k, then either X has rank k, or del f(X) = 0.


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