Non-Canonical Hamiltonian Monte Carlo

• URL: http://arxiv.org/abs/2008.08191v1
• Date: Tue, 18 Aug 2020 23:25:20 GMT
• Title: Non-Canonical Hamiltonian Monte Carlo
• Abstract summary: Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. We present a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures.
• Score: 7.6146285961466
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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