Intrinsic effective sample size for manifold-valued Markov chain Monte Carlo via kernel discrepancy
Authors: Kisung You
Organizations: Department of Mathematics, Baruch College · Department of Mathematics, The Graduate Center, City University of New York
Abstract
Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise effective sample sizes can change under rotations, chart changes, or alternative embeddings of the same underlying path. We propose an intrinsic effective sample size based on kernel discrepancy. The proposed quantity is the number of independent draws that would yield the same expected squared kernel discrepancy between the empirical distribution and the target distribution. This gives an exact finite-sample risk interpretation, an asymptotic integrated-autocorrelation representation, and a coordinate-free diagnostic whenever the kernel respects the geometry of the state space. We establish invariance under transported kernels, operator and principal-direction interpretations, and consistency of a lag-window estimator under boundedness and absolute-regularity conditions. We also discuss valid kernel constructions on manifolds, emphasizing that geodesic Gaussian kernels are not generally positive definite on curved spaces. Sphere experiments illustrate rotation invariance and calibration of the proposed diagnostic against empirical distributional error.
Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two Rényi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effective sample size that discounts nearby or duplicate particles while matching ordinary effective sample size for well-separated particles. We prove monotonicity, small- and large-scale asymptotics, deterministic-weight consistency, and a bounded-ratio self-normalized importance-sampling extension for compact manifolds without boundary. On spheres, the unlogged profile decomposes into spherical-harmonic energies that recover mean-direction, von Mises-Fisher-type, and Bingham-type summaries. Sphere-based experiments show that the profile reveals antipodal, girdle, multimodal, and duplicate-particle structure missed by weight-only and first-moment spherical summaries.
We develop sampling methods, which consist of Gaussian invariant versions of random walk Metropolis (RWM), Metropolis adjusted Langevin algorithm (MALA) and second order Hessian or Manifold MALA. Unlike standard RWM and MALA, we show that Gaussian invariant sampling can lead to ergodic estimators with improved statistical efficiency. This is due to a remarkable property of Gaussian invariance that allows us to obtain exact analytical solutions to the Poisson equation for Gaussian targets. These solutions can be used to construct efficient and easy to use control variates for variance reduction of estimators under any intractable target. We demonstrate the new samplers and estimators in several examples, including high dimensional targets in latent Gaussian models where we compare against several advanced methods and obtain state-of-the-art results. We also provide theoretical results regarding geometric ergodicity, and an optimal scaling analysis that shows the dependence of the optimal acceptance rate on the Gaussianity of the target.
Michalis K. Titsias, Angelos Alexopoulos, Siran Liu +1
Sequential Monte Carlo (SMC) methods are a natural tool for post-hoc conditioning of pretrained generative models, but in many applications the mutation kernels used by the particle system are biased approximations of an ideal Feynman--Kac flow. This paper develops a non-asymptotic error analysis for such SMC samplers. Under forward-smoothing forgetting conditions, we decompose the total error into a kernel bias, measuring the effect of replacing the ideal transition kernels by approximate ones, and a finite-particle Monte Carlo error. Our approach relies on extending local Doeblin-type conditions and Lyapunov drift arguments for Markov kernels to conditional distributions, thereby enabling a principled control of the bias. We then instantiate this general framework for conditional sampling with score-based diffusion models, and derive the first non-asymptotic error bound that jointly controls initialization error, time discretization, and score approximation in the reverse diffusion dynamics as well as finite-particle Monte Carlo error.
Stanislas Strasman, Gabriel Victorino Cardoso, Sylvain Le Corff +2