Integral geometry, Hamiltonian dynamics, and Markov Chain Monte Carlo

Abstract

This thesis presents applications of differential geometry and graph theory to the design and analysis of Markov chain Monte Carlo (MCMC) algorithms. MCMC algorithms are used to generate samples from an arbitrary probability density [pi] in computationally demanding situations, since their mixing times need not grow exponentially with the dimension of [pi]. However, if [pi] has many modes, MCMC algorithms may still have very long mixing times. It is therefore crucial to understand and reduce MCMC mixing times, and there is currently a need for global mixing time bounds as well as algorithms that mix quickly for multi-modal densities. In the Gibbs sampling MCMC algorithm, the variance in the size of modes intersected by the algorithm's search-subspaces can grow exponentially in the dimension, greatly increasing the mixing time. We use integral geometry, together with the Hessian of r and the Chern-Gauss-Bonnet theorem, to correct these distortions and avoid this exponential increase in the mixing time. Towards this end, we prove a generalization of the classical Crofton's formula in integral geometry that can allow one to greatly reduce the variance of Crofton's formula without introducing a bias. Hamiltonian Monte Carlo (HMC) algorithms are some the most widely-used MCMC algorithms. We use the symplectic properties of Hamiltonians to prove global Cheeger-type lower bounds for the mixing times of HMC algorithms, including Riemannian Manifold HMC as well as No-U-Turn HMC, the workhorse of the popular Bayesian software package Stan. One consequence of our work is the impossibility of energy-conserving Hamiltonian Markov chains to search for far-apart sub-Gaussian modes in polynomial time. We then prove another generalization of Crofton's formula that applies to Hamiltonian trajectories, and use our generalized Crofton formula to improve the convergence speed of HMC-based integration on manifolds. We also present a generalization of the Hopf fibration acting on arbitrary- ghost-valued random variables. For [beta] = 4, the geometry of the Hopf fibration is encoded by the quaternions; we investigate the extent to which the elegant properties of this encoding are preserved when one replaces quaternions with general [beta] > 0 ghosts.