Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling

We study sampling from π(x)proptoe-V(x) on ℝd, where V is α-strongly convex and β-smooth with condition number kappa=β/α. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce textitreflected proximal dynamics, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant chi2 divergence, both methods achieve total variation error varepsilon using widetilde O(√kappad1/4) expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields widetilde O(kappa6/7d1/4) expected oracle queries. In both regimes, the dependence on the target accuracy varepsilon is polylog(1/varepsilon), making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.