arXiv subjects

Alessandro Barp

Publications and source records attributed to Alessandro Barp.

22 records · Page 2Linked to original sources

arXiv2019

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

In this paper we show that the Hamiltonian Monte Carlo method for compact Lie groups constructed in \cite{kennedy88b} 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 $\G\times \g$ we recover the Euler-Arnold formulation of geodesic motion, and construct explicit HMC schemes that extend \cite{kennedy88b,Kennedy:2012} 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.

math.DG↗

arXiv2018

Posterior Integration on a Riemannian Manifold

The geodesic Markov chain Monte Carlo method and its variants enable computation of integrals with respect to a posterior supported on a manifold. However, for regular integrals, the convergence rate of the ergodic average will be sub-optimal. To fill this gap, this paper extends the efficient posterior integration method of Oates et al. (2017) to the case of a Riemannian manifold. In contrast to the original Euclidean case, no non-trivial boundary conditions are needed for a closed manifold. The method is assessed through simulation and deployed to compute posterior integrals for an Australian Mesozoic paleomagnetic pole model, whose parameters are constrained to lie on the manifold $M = \mathbb{S}^2 \times \mathbb{R}_+$.

stat.ME↗

arXiv2017

Geometry and Dynamics for Markov Chain Monte Carlo

Markov Chain Monte Carlo methods have revolutionised mathematical computation and enabled statistical inference within many previously intractable models. In this context, Hamiltonian dynamics have been proposed as an efficient way of building chains which can explore probability densities efficiently. The method emerges from physics and geometry and these links have been extensively studied by a series of authors through the last thirty years. However, there is currently a gap between the intuitions and knowledge of users of the methodology and our deep understanding of these theoretical foundations. The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners and other users of the methodology with only a basic understanding of Monte Carlo methods. This will be complemented with some discussion of the most recent advances in the field which we believe will become increasingly relevant to applied scientists.

stat.CO↗

arXiv2015

A numerical study of the 3D random interchange and random loop models

We have studied numerically the random interchange model and related loop models on the three-dimensional cubic lattice. We have determined the transition time for the occurrence of long loops. The joint distribution of the lengths of long loops is Poisson-Dirichlet with parameter 1 or 1/2.

math-ph↗