2020/08/18 by James A. Brofos, Brofos, James A., Roy R. Lederman +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Model Reduction and Neural Networks
paper · pdf · doi:10.48550/arxiv.2008.08191
openalex publication_date 2020/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. Numerical integrators designed for the canonical structure are incompatible with motion generated by non-canonical dynamics. These non-canonical dynamics, motivated by examples in physics and symplectic geometry, correspond to techniques such as preconditioning which are routinely used to improve algorithmic performance. Indeed, recently, a special case of non-canonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with non-canonical structure. To summarize our contributions: (i) we develop non-canonical HMC from foundations in symplectic geomtry; (ii) we construct an HMC procedure using implicit integration that satisfies the detailed balance; (iii) we propose to accelerate the sampling using an \em approximate explicit methodology; (iv) we study two novel, randomly-generated non-canonical structures: magnetic momentum and the coupled magnet structure, with implicit and explicit integration.