2014/02/21 by Schneider, Reinhold, Uschmajew, André · 3 citations
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1402.5284
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 Mk 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 Mk. The pointwise convergence is obtained for real-analytic functions on the basis of a Łojasiewicz 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 Mk, 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 Mk: if X is a critical point of f on M≤ k, then either X has rank k, or ∇ f(X) = 0.