Trust-Region Methods on Riemannian Manifolds

Abstract

A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.

Riemannian BFGS Algorithm with Applications

“…Optimization on manifolds finds applications in two broad classes of situations: classical equality-constrained optimization problems where the constraints specify a submanifold of R n ; and problems where the objective function has continuous invariance properties that we want to eliminate for various reasons, e.g., efficiency, consistency, applicability of certain convergence results, and avoiding failure of certain algorithms due to degeneracy. As a result, the generalization to manifolds of algorithms for unconstrained optimization in R n can yield useful and efficient numerical methods; see, e.g., recent work on Riemannian trust-region methods [2] and other methods mentioned in [3]. Since BFGS is one of the classical methods for unconstrained optimization (see [7,10]), it is natural that its generalization be a topic of interest.…”

An Implicit Riemannian Trust-Region Method for the Symmetric Generalized Eigenproblem

“…The minimization of the models is done via an iterative process, which is referred to as the inner iteration, to distinguish it with the principal outer iteration. We present here the process in a way that does not require a background in differential geometry; we refer to [13] for the mathematical foundations of the technique.…”

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives

“…Methods of Riemannian Optimization (RO) have been recently a subject of great interest in data analytics communities; see, for example, [Absil et al, 2007, 2008, Bento et al, 2016, Bonnabel, 2013, Cambier and Absil, 2016. Some optimization problems discussed in previous sections, can be naturally formulated on Riemannian manifolds, so as to directly benefit from the underlying geometric structures that can be exploited to significantly reduce the cost of obtaining solutions.…”