On the Singular Value Manifold and Numerical Stabilization of Algorithms with Orthogonality Constraints
S.C. Douglas · 2006
Recently, interest has risen in adaptive algorithms that implicitly impose orthogonality constraints on an adjustable matrix. In practice, parameter deviations from orthogonality can occur due to a chosen algorithm's numerical implementation. This paper introduces the geometry of and adaptive algorithms for the singular value manifold to mitigate these numerical effects. Both gradient and Newton-based methods on the singular value manifold are derived. Applications to single-step and iterative orthogonalization reveal relationships between existing orthogonalization methods as well as novel, fast-converging approximate Newton procedures for this task. Simulations are used to explore their performances