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 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 M it intersects. This variance can grow exponentially with the search-subspace dimension, greatly slowing the algorithm. Eliminating this variance allows us to use search-subspaces of many times greater dimension, allowing us to sample very rare events that a lower-dimensional search-subspace would be unlikely to intersect. To extend this method to events rare for reasons other than M having lower dimension, we prove a new theorem in integral geometry that uses the Chern-Gauss-Bonnet theorem’s curvature form to reweight sampling probabilities. Finally, we demonstrate the computational effectiveness and speedup of our method by numerically applying it to a random matrix sampling problem.