A Differential Geometric Approach To Eigenspace Computation

Pierre-Antoine Absil

Florida State

November 17, 4:15pm


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 (i.e.
subspaces) and the matrix representations of these entities.  These matrix
representations of subspaces admit a topological structure of GL-principal
fiber bundle over the Grassmann manifold.  Moreover, with a suitable
choice of metrics, the bundle projection can be turned into a Riemannian

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).

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