Effect of including the divergence correction term on mixing time

Determine how incorporating the divergence correction term (∇·M^{-1}) into the proposal (moving from the Preconditioned Langevin Algorithm to the Weighted Langevin Algorithm) affects the mixing-time guarantees of Metropolis-adjusted schemes for constrained sampling with self-concordant metrics, and characterize the resulting dependence on dimension and curvature parameters.

Background

MAPLA omits the (∇·M^{-1}) term present in the weighted Langevin dynamics and ManifoldMALA for computational simplicity. The authors highlight the open question of the theoretical impact of adding this term on mixing behavior, beyond practical implementation concerns.

Clarifying this effect would connect MAPLA’s analysis to ManifoldMALA and shed light on whether the divergence correction improves, worsens, or leaves unchanged the asymptotic and non-asymptotic mixing properties under the self-concordant metric framework.