Riemannian Optimization Method on the Flag Manifold for Independent Subspace Analysis

Recent authors have investigated the use of manifolds and Lie group methods for independent component analysis (ICA), including the Stiefel and the Grassmann manifolds and the orthogonal group O(n). In this paper we introduce a new class of manifold, the generalized flag manifold, which is suitable for independent subspace analysis. The generalized flag manifold is a set of subspaces which are orthogonal to each other, and includes the Stiefel and the Grassmann manifolds as special cases. We describe how the independent subspace analysis problem can be tackled as an optimization on the generalized flag manifold. We propose a Riemannian optimization method on the generalized flag manifold by adapting an existing geodesic formula for the Stiefel manifold, and present a new learning algorithm for independent subspace analysis based on this approach. Experiments confirm the effectiveness of our method.