Riemannian Optimization Method on Generalized Flag Manifolds for Complex and Subspace ICA
Yasunori Nishimori, Shotaro Akaho, Mark D. Plumbley · AIP conference proceedings · 2006
In this paper we introduce a new class of manifolds, generalized flag manifolds, for the complex and subspace ICA problems. A generalized flag manifold is a manifold consisting of subspaces which are orthogonal to each other. The class of generalized flag manifolds include the class of Grassmann manifolds. We extend the Riemannian optimization method to include this new class of manifolds by deriving the formulas for the natural gradient and geodesics on these manifolds. We show how the complex and subspace ICA problems can be solved by optimization of cost functions on a generalized flag manifold. Computer simulations demonstrate our algorithm gives good performance compared with the ordinary gradient descent method.