Momentum Maps and Stochastic Clebsch Action Principles

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity ma...

Momentum Maps and

Momentum Maps and Morita Equivalence

We introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including ordinary Hamiltonian G-spaces, Lu’s momentum maps of Poisson group actions, and the group-valued momentum maps of Alekseev–Malkin–Meinrenken. More precisely, we carry out the following program: (1) We de...

Momentum and Optimal Stochastic Search

The rate of convergence for gradient descent algorithms, both batch and stochastic, can be improved by including in the weight update a “momentum” term proportional to the previous weight update. Several authors [1, 2] give conditions for convergence of the mean and covariance of the weight vector for momentum LMS with constant learning rate. However stochastic algorithms require that the learn...

Rigidity of Infinitesimal Momentum Maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.