Neural learning by geometric integration of reduced 'rigid-body' equations
Journal of Computational and Applied Mathematics
Learning independent components on the orthogonal group of matrices by retractions
Neural Processing Letters
Numerical methods for ordinary differential equations on matrix manifolds
Journal of Computational and Applied Mathematics
Lie-group-type neural system learning by manifold retractions
Neural Networks
Descent methods for optimization on homogeneous manifolds
Mathematics and Computers in Simulation
Preserving poisson structure and orthogonality in numerical integration of differential equations
Computers & Mathematics with Applications
An algorithm to compute averages on matrix Lie groups
IEEE Transactions on Signal Processing
An introduction to Lie group integrators - basics, new developments and applications
Journal of Computational Physics
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Numerical integration of ODEs on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n × k matrices with orthonormal columns, of particular interest is the case when $n\gg k$. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order nk2. Second, they should be intrinsic in the sense that they require only the ODE vector field to be defined on the Stiefel manifold, as opposed to, for instance, projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the QR decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.