Riemannian Optimization Method on the Flag Manifold for Independent Subspace Analysis
نویسندگان
چکیده
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.
منابع مشابه
Geodesic Learning Algorithms Over Flag Manifolds
Recently manifold structures have attracted attentions in two folds in the machine learning literature. One is in the manifold learning problem, that is learning the intrinsic manifold structure in high dimensional datasets. Another is in the information geometric approach to learning – exploiting the geometry of the parameter space of learning machines such as neural networks for improving con...
متن کاملNatural Conjugate Gradient on Complex Flag Manifolds for Complex Independent Subspace Analysis
We study the problem of complex-valued independent subspace analysis (ISA). We introduce complex flag manifolds to tackle this problem, and, based on Riemannian geometry, propose the natural conjugate gradient method on this class of manifolds. Numerical experiments demonstrate that the natural conjugate gradient method yields better convergence compared to the natural gradient geodesic search ...
متن کاملON THE LIFTS OF SEMI-RIEMANNIAN METRICS
In this paper, we extend Sasaki metric for tangent bundle of a Riemannian manifold and Sasaki-Mok metric for the frame bundle of a Riemannian manifold [I] to the case of a semi-Riemannian vector bundle over a semi- Riemannian manifold. In fact, if E is a semi-Riemannian vector bundle over a semi-Riemannian manifold M, then by using an arbitrary (linear) connection on E, we can make E, as a...
متن کاملOn a class of paracontact Riemannian manifold
We classify the paracontact Riemannian manifolds that their Riemannian curvature satisfies in the certain condition and we show that this classification is hold for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X, ξ).S = 0, where S is the Ricci tensor.
متن کاملGeometric Optimization Methods for Adaptive Filtering
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 generalize...
متن کامل