Magnetic Manifold Hamiltonian Monte Carlo
This work addresses sampling challenges in constrained domains, such as on spheres, for applications in statistics and machine learning, representing an incremental extension of magnetic HMC to manifolds.
The paper tackles the problem of sampling from distributions restricted to constrained sets, such as embedded manifolds, by introducing magnetic manifold Hamiltonian Monte Carlo (HMC), which improves sampling performance compared to canonical manifold-constrained HMC.
Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.