Optimal scaling and curvature correction for constrained MCMC on spheres

Abstract

Sampling Gaussian measures restricted to nonlinear manifolds in high dimensions requires a proposal kernel that respects both geometric constraints and dimensional scaling requirements. We introduce tangent preconditioned Crank-Nicolson (pCN), which combines a manifold projection with a cancellation of the ambient Gaussian contribution, leaving only a geometric residual. Considering a shifted sphere with $\sqrt d$ scaling of radius and centre as a tractable benchmark embedded manifold, we prove that uncorrected tangent pCN results in a proposal noise scale of order $d^{-1/6}$ (compared to $d^{-1/2}$ for a tangent random-walk). We establish asymptotic optimal tuning for the geodesic expected squared jump distance which is maximised at an acceptance probability of 0.574 for the family of drift-corrected tangent proposals containing tangent pCN and tangent MALA. We also prove a joint diffusion limit for the parallel and orthogonal coordinates to the sphere's centre vector of tangent pCN. The orthogonal diffusion speed matches the limiting geodesic expected squared jump distance and is maximised at acceptance probability 0.574, whereas the parallel speed is maximised at 0.712, demonstrating that optimal tuning depends on the observable. We then identify an intrinsic precision correction characterised by potential Hessian and Ricci curvature that yields a proposal kernel with dimension-independent fixed step size limit. We discuss a generalisation to arbitrary hypersurfaces and present nondegenerate scaling of tangent pCN applied to conditioned diffusions under resolution refinement.