Lagrangian Manifold Monte Carlo on Monge Patches
This addresses the slow performance of existing geometric MCMC methods for practitioners dealing with high-dimensional statistical inference, though it appears incremental as an alternative metric within an established framework.
The paper tackles the computational inefficiency of geometric MCMC methods for distributions with varying curvature by proposing a new Riemannian metric based on Monge patch embeddings, which reduces iteration complexity from cubic to quadratic in dimensionality while using only first-order gradients.
The efficiency of Markov Chain Monte Carlo (MCMC) depends on how the underlying geometry of the problem is taken into account. For distributions with strongly varying curvature, Riemannian metrics help in efficient exploration of the target distribution. Unfortunately, they have significant computational overhead due to e.g. repeated inversion of the metric tensor, and current geometric MCMC methods using the Fisher information matrix to induce the manifold are in practice slow. We propose a new alternative Riemannian metric for MCMC, by embedding the target distribution into a higher-dimensional Euclidean space as a Monge patch and using the induced metric determined by direct geometric reasoning. Our metric only requires first-order gradient information and has fast inverse and determinants, and allows reducing the computational complexity of individual iterations from cubic to quadratic in the problem dimensionality. We demonstrate how Lagrangian Monte Carlo in this metric efficiently explores the target distributions.