Hamiltonian Monte Carlo Method and Symplectic Geometry

Hamiltonian Monte Carlo (HMC) method is an application of non-Euclidean geometry to inverse problems. It is a probabilistic sampling method with the basis of Hamiltonian dynamics. One of the main advantages of the HMC algorithm is to draw independent samples from the state space with a higher acceptance rate than other MCMC methods. In order to understand how a higher acceptance rate is achieved, I have studied HMC in the light of symplectic geometry. Hamiltonian dynamics is defined on the phase space (cotangent bundle), which has a natural symplectic structure, i.e. a differential two-form that is non-degenerate and closed.

Symplectic geometry lies at the very foundations of physics: Geometry is the method of abstracting the solutions of physical phenomena. Once the use of phase space in the solutions of mechanical systems (e.g. simple harmonic motion, or ray-tracing) is abstracted via geometry, then it can be used in other branches such as optimization problems (e.g. Hamiltonian Monte Carlo). I present two different applications of symplectic geometry: Ray-tracing and Hamiltonian Monte Carlo.

First, the Hamiltonian function is defined on the phase space, which corresponds to an invariant of the system (e.g. total energy for the HMC method and wavefront normal for ray-tracing problem), and then by using the non-degeneracy property, a vector field can be found in which Hamiltonian function is invariant along the integral curves of the field. The invariance of the Hamiltonian function results in a high acceptance rate, where we apply the accept-reject test to satisfy the detailed-balance property.

After describing the concept of phase space for both mechanical systems and optimization problems, I am going to show different applications of HMC, including 2-dimensional travel-time tomography on a synthetic complex velocity structure.