Pierre-Antoine Absil
Department of Electrical Engineering and Computer Science
Institut Montefiori
University of Liege
Invariant Subspace Computation - A Geometric Approach
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 n-by-n data matrix. This problem nicely lends itself to a geometric approach, since the iterates belong to a particular non-Euclidean set, the set of fixed-dimensional 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 (subspaces) and the numerical representations of these entities. It turns out that matrix representations of subspaces admit a topological structure of GL-principal fiber bundle over the Grassmann manifold. Interestingly, a similar structure appears in several other problems, including linear systems realization. In this talk, we will present two recently proposed numerical algorithms that stem from this Grassmannian approach. The first one is 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. The second algorithm is a generalization to the Grassmann manifold of the classical Rayleigh quotient iteration. The new iteration converges locally cubically to the eigenspaces of a symmetric data matrix, and a two-sided version is proposed for tackling the nonsymmetric case. This is work was done in collaboration with Rodolphe Sepulchre (University of Liege, Belgium), Robert Mahony (Australian National University) and Paul Van Dooren (Universite Catholique de Louvain, Belgium).