On the Design of Gradient Algorithms Employing Orthogonal Matrix Constraints

Algorithms for adapting orthogonal matrices in optimization and signal processing typically employ the geometry of either the Grassmann manifold or the Stiefel manifold depending on the chosen cost function. In this paper, we develop gradient adaptive algorithms that use the geometry of both manifolds in their operation. Such algorithms offer a straightforward way to mitigate numerical error accumulation due to discretization of the coefficient updates. Examples drawn from subspace tracking and eigenvector analysis illustrate the usefulness of the design methods.