Hamiltonian Monte Carlo on Lie Groups and Constrained Mechanics on Homogeneous Manifolds

Abstract

In this paper we show that the Hamiltonian Monte Carlo method for compact Lie groups constructed in [18] using a symplectic structure can be recovered from canonical geometric mechanics with a bi-invariant metric. Hence we obtain the correspondence between the various formulations of Hamiltonian mechanics on Lie groups, and their induced HMC algorithms. Working on G × g we recover the Euler-Arnold formulation of geodesic motion, and construct explicit HMC schemes that extend [18,19] to non-compact Lie groups by choosing metrics with appropriate invariances. Finally we explain how mechanics on homogeneous spaces can be formulated as a constrained system over their associated Lie groups, and how in some important cases the constraints can be naturally handled by the symmetries of the Hamiltonian.

Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents

“…In general, it is desirable to use a Riemannian metric that reflects the intrinsic symmetries of the sample space, mathematically described by a Lie group action. Indeed, by using an invariant Riemannian metric, one greatly simplifies the equations of motion of the geodesic flow, reducing the usual 2nd order Euler-Langrange equations to the 1st order Euler-Arnold equations [144][145][146], with tractable solutions in many cases of interest, e.g., for naturally reductive homogeneous spaces; including R d , the space of positive definite matrices, Stiefel manifolds, Grassmannian manifolds, and many Lie groups. In such cases, it is possible to find a Riemannian metric whose geodesic flow is known and given by the Lie group exponential [16,147,148].…”

Curved schemes for stochastic differential equations on, or near, manifolds

“…Ours is a related but distinct approach that does not require symplecticity. Various authors (Barp [38], Laurent & Vilmart [39], Zappa et al [40]) consider the question of Monte Carlo on a manifold that has been specified explicitly, but below we concentrate on the case where the manifold is only given approximately via the typical set.…”

Irreversible Langevin MCMC on Lie Groups

“…Vector fields tangent to T G (i.e., elements of Γ (T T G)) can be expanded in terms of left-invariant vector fields e i and the fibre-coordinate vector fields ∂ v i , (i.e., Γ (T T G) ∼ = Γ (T G ⊕ T g)). We consider noise Hamiltonians that depend only on position, H i = U i • π where U i : G → R, so the corresponding Hamiltonian vector fields can be written as X Hi = −e j (U i )∂ v j , (see [3]). Hence the stochastic process (1) on T G can be split up into a Langevin part…”

“…The phase transitions of this system were analysed using a sampling method [2]. In lattice gauge theory one typically uses the HMC algorithm for semi-simple compact Lie groups which was originally presented in [16] and extended to arbitrary Lie groups in [3], see also [11,12,17]. In [5], it was shown how to construct HMC on homogeneous manifolds using symplectic reduction, which includes sampling on Lie groups as a special case.…”