Riemannian BFGS Algorithm with Applications
نویسندگان
چکیده
In this paper, we present a retraction-based Riemannian BFGS approach (RBFGS). Of particular interest is the choice of transport used to move information between tangent spaces and the different ways of implementing the RBFGS algorithm. We consider parallel translation along a geodesic and vector transport by projection on the unit sphere and the compact Stiefel manifold.
منابع مشابه
Properties of the BFGS method on Riemannian manifolds
We discuss the BFGS method on Riemannian manifolds and put a special focus on the degrees of freedom which are offered by this generalization. Furthermore, we give an analysis of the BFGS method on Riemannian manifolds that are isometric toRn.
متن کاملA Riemannian quasi-Newton method for computing the Karcher mean of symmetric positive definite matrices
This paper tackles the problem of computing the Karcher mean of a collection of symmetric positive-definite matrices. We present a concrete limited-memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects on the manifold of symmetric positive-definite matrices that are required for Riemannian opt...
متن کاملGeometric Optimization in Machine Learning
Machine learning models often rely on sparsity, low-rank, orthogonality, correlation, or graphical structure. The structure of interest in this chapter is geometric, specifically the manifold of positive definite (PD) matrices. Though these matrices recur throughout the applied sciences, our focus is on more recent developments in machine learning and optimization. In particular, we study (i) m...
متن کاملOptimization Methods on Riemannian Manifolds and Their Application to Shape Space
We extend the scope of analysis for linesearch optimization algorithms on (possibly infinitedimensional) Riemannian manifolds to the convergence analysis of the BFGS quasi-Newton scheme and the Fletcher–Reeves conjugate gradient iteration. Numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods.
متن کاملA Broyden Class of Quasi-Newton Methods for Riemannian Optimization
This paper develops and analyzes a generalization of the Broyden class of quasiNewton methods to the problem of minimizing a smooth objective function f on a Riemannian manifold. A condition on vector transport and retraction that guarantees convergence and facilitates efficient computation is derived. Experimental evidence is presented demonstrating the value of the extension to the Riemannian...
متن کامل