Averaging Complex Subspaces via a Karcher Mean Approach

TL;DR

This paper introduces a conjugate gradient optimization method for computing the Karcher mean on complex Grassmannian manifolds, enabling efficient averaging of complex subspaces with applications in blind identification.

Contribution

It presents a novel intrinsic conjugate gradient algorithm tailored for the complex Grassmannian, including geometric definitions and an efficient step-size strategy for one-dimensional subspaces.

Findings

Effective computation of Karcher mean demonstrated on complex subspaces.

Method improves efficiency over existing approaches.

Numerical experiments validate applicability to blind identification.

Abstract

We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometric concepts required for an intrinsic conjugate gradient method. In particular, proper definitions of geodesics, parallel transport, and the Riemannian gradient of the Karcher mean function are presented. We provide an efficient step-size selection for the special case of one dimensional complex subspaces and illustrate how the method can be employed for blind identification via numerical experiments.