Reflective Hamiltonian Monte Carlo: Mixing Analysis and Application to Sampling on Stiefel Manifold

Sampling from distributions with bounded supports is a fundamental challenge in constrained statistical inference. Reflective Hamiltonian Monte Carlo (ReHMC) provides a useful sampling approach for this setting. However, it relies on convexity assumptions on the support and lacks non-asymptotic theoretical guarantees such as mixing-time bounds. To bridge this gap, we propose a convex-container plus thinning framework that is applicable to arbitrary target densities with bounded support. We establish the first non-asymptotic total-variation mixing-time bounds for ReHMC, achieving a polynomial dimension dependence of $O(d^2)$ for $L$-smooth targets, though with exponential dependence on smoothness parameters. Under an additional $m$-strong convexity assumption, we derive a sharper bound that eliminates this exponential dependence. We further apply this approach to sampling on the Stiefel manifold via a well-conditioned polar reparameterization and demonstrate improved numerical stability and sampling efficiency in simulation studies.

Sampling from probability distributions is a central task in statistics and machine learning, but it becomes difficult when the samples must satisfy constraints. For example, some models require parameters to stay inside a bounded region or to form an orthonormal matrix. This paper studies a reflective sampling method that keeps samples inside the allowed space by making them bounce off boundaries. We introduce a new framework that extends this method to more general bounded supports and proves guarantees on how fast it reaches the desired distribution. We also show how the method can be used for sampling on the Stiefel manifold, a geometric space that appears in many constrained learning problems. The proposed approach provides a more stable and efficient tool for constrained statistical computation.