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The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression. [ABSTRACT FROM AUTHOR]

Proposes a generalization of the classical Rayleigh quotient iteration (RQI) which computes p-dimensional invariant subspace of A. Review of the classical RQI; Representation of p-dimensional subspaces by means of matrices and an associated measure of distance between subspaces; Convergence of the Grassmann-RQI; Comparison with other Grassmannian methods.

In this paper the effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semiglobal stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback. [ABSTRACT FROM AUTHOR]