Integral geometry for Markov chain Monte Carlo: overcoming the curse of search-subspace dimensionality

TL;DR

This paper introduces an integral geometry-based reweighting method for Markov chain Monte Carlo algorithms, significantly improving their efficiency in high-dimensional sampling, especially for rare events, by analytically reducing variance caused by search-subspace orientation.

Contribution

It develops a novel curvature-based reweighting technique using integral geometry to enhance MCMC convergence and extend sampling capabilities to higher-dimensional and rare event scenarios.

Findings

Achieves faster convergence in high-dimensional MCMC sampling.

Enables sampling of rare events with larger search-subspaces.

Provides new theoretical bounds for real algebraic manifold volumes.

Abstract

We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold $\mathcal{M}$. Our method exploits the symmetries of the algorithms' isotropic random search-direction subspaces to analytically average out the variance in the intersection volume caused by the orientation of the search-subspace with respect to the manifold $\mathcal{M}$ it intersects. This variance can grow exponentially with the dimension of the search-subspace, greatly slowing down the algorithm. Eliminating this variance allows us to use search-subspaces of dimensions many times greater than would otherwise be possible, allowing us to sample very rare events that a lower-dimensional search-subspace would be unlikely to intersect. To extend this method to events that are rare for reasons other than their support $\mathcal{M}$ having a lower dimension, we formulate and prove a new theorem in integral geometry that makes use of the curvature form of the Chern-Gauss-Bonnet theorem to reweight sampling probabilities. On the side, we also apply our theorem to obtain new theoretical bounds for the volumes of real algebraic manifolds. Finally, we demonstrate the computational effectiveness and speedup of our method by numerically applying it to the conditional stochastic Airy operator sampling problem in random matrix theory.