ABSTRACT


The theme of this presentation is eigenspace computation, an ubiquitous task in engineering and physical sciences. More precisely, we consider the problem of iteratively refining estimates of an eigenspace of an nbyn data matrix. This problem nicely lends itself to a geometric approach, since the iterates belong to a particular nonEuclidean set, the set of fixeddimensional subspaces of Rn, commonly referred to as the Grassmann manifold. In order to derive numerical algorithms from the geometric approach, it is essential to understand the relation between the geometric objects (i.e. subspaces) and the matrix representations of these entities. These matrix representations of subspaces admit a topological structure of GLprincipal fiber bundle over the Grassmann manifold. Moreover, with a suitable choice of metrics, the bundle projection can be turned into a Riemannian submersion. We take advantage of this geometric structure to derive a Newton method that relies on the Riemannian geometry of the Grassmann manifold. It converges locally quadratically to the eigenspaces of the data matrix, and even cubically when the data matrix is symmetric. This method is closely related to several available algorithms for eigenspace computation.This presentation builds on collaborations with Rodolphe Sepulchre (University of Liege, Belgium), Robert Mahony (Australian National University), Paul Van Dooren (Universite Catholique de Louvain, Belgium) and Kyle Gallivan (Florida State University). Return to Applied Math Colloquium home page 