Newton-like methods for numerical optimization on manifolds

• Published in: Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004 Vol. 1; pp. 136 - 139 Vol.1
• Main Authors: Huper, K., Trumpf, J.
• Format: Conference Proceeding
• Language: English
• Published: Piscataway NJ IEEE 2004
• Subjects: Applied sciences; Australia Council; Biomedical signal processing; Constraint optimization; Convergence; Cost function; Exact sciences and technology; Information, signal and communications theory; Manifolds; Mathematical methods; Optimization methods; Signal processing algorithms; Subspace constraints; Systems engineering and theory; Telecommunications and information theory; Symmetric matrix; Subspace method; Signal processing; Cost function; Numerical method; Parameterization; Algorithm; Newton method; Constrained optimization
• ISBN: 0780386221; 9780780386228
• DOI: 10.1109/ACSSC.2004.1399106

Many problems in signal processing require the numerical optimization of a cost function, which is defined on a smooth manifold. Especially, orthogonally or unitarily constrained optimization problems tend to occur in signal processing tasks involving subspaces. In this paper we consider Newton-like methods for solving these types of problems. Under the assumption that the parameterization of the manifold is linked to so-called Riemannian normal coordinates our algorithms can be considered as intrinsic Newton methods. Moreover, if there is not such a relationship, we still can prove local quadratic convergence to a critical point of the cost function by means of analysis on manifolds. Our approach is demonstrated by a detailed example, i.e., computing the dominant eigenspace of a real symmetric matrix.