Geometric Optimization Methods for Adaptive Filtering

TL;DR

This thesis introduces new geometric optimization algorithms on Riemannian manifolds, specifically for adaptive filtering and subspace tracking, with proven convergence properties and applications to eigenvalue problems.

Contribution

It generalizes classical Euclidean optimization techniques to Riemannian manifolds and develops two new algorithms with quadratic and superlinear convergence.

Findings

Algorithms demonstrate quadratic and superlinear convergence.

Effective application to eigenvalue and singular value problems.

Provides a new perspective on constrained optimization on manifolds.

Abstract

The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization problem. Some classical optimization techniques on Euclidean space are generalized to Riemannian manifolds. Several algorithms are presented and their convergence properties are analyzed employing the Riemannian structure of the manifold. Specifically, two new algorithms, which can be thought of as Newton's method and the conjugate gradient method on Riemannian manifolds, are presented and shown to possess quadratic and superlinear convergence, respectively. These methods are applied to several eigenvalue and singular value problems, which are posed as constrained optimization problems. ...