Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality

Abstract

The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety M_ k of real m n 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 ojasiewicz 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 f(X) = 0.

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