Extend exponential-integrator error bounds to splitting discretizations on manifolds

Establish discretization error bounds for two-stage splitting schemes for Langevin-type dynamics on manifolds (including kinetic Langevin on Lie groups) that are analogous to the existing exponential-integrator or geometric Euler–Maruyama bounds, thereby enabling rigorous nonasymptotic analyses of such structure-preserving discretizations.

Background

The paper uses a two-stage operator splitting discretization to preserve Lie group structure, contrasting with more commonly analyzed exponential-integrator or geometric Euler–Maruyama schemes. While prior work has quantified discretization errors for manifold Langevin (without momentum), extending those bounds to splitting schemes remains technically challenging.

The authors note that existing tools tailored to exponential-integrator-based analysis have difficulties applying to their splitting method, highlighting a gap in discretization error theory for structure-preserving schemes on manifolds.

References

How to expand the bound for numerical error for exponential integrator in their work to splitting discretization is still unclear.

— Convergence of Kinetic Langevin Monte Carlo on Lie groups  (2403.12012 - Kong et al., 2024) in Section "More details about Sec. \ref{sec_convergence_splitting}", Remark "Comparison with \citep{cheng2022efficient}"

Although we do not prove this directly, the sampler is indeed empirically first-order stationary with respect to the exact marginals themselves. For the comparison with Figure~\ref{fig:gmm1d}(c) we evaluate $\frac1T\int_0T(_t\,|\,q_t) t$ against the analytic scores along the frozen-score SDE interpolation of the scheme, the kinetic analog of eq:numerical-sde. Its gap to the exact-flow value decays like $O(h)$ (Figure~\ref{fig:gmm1d-app}(d)).

— First-Order Stationarity of Reverse Diffusions  (2609.31612 - Chen et al., 25 Sep 2026) in Appendix, Section “Numerical experiments: details,” subsection “Critically damped flow”