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Convergence of mutual information along underdamped Langevin and other Markov chains

Establish the rate at which the mutual information I(X0; Xt) decreases along other Markov chains beyond those analyzed in the paper, in particular along the underdamped (kinetic) Langevin dynamics.

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Background

The work analyzes mutual information decay for the overdamped Langevin diffusion and its discretization (ULA). Many practical sampling algorithms use other dynamics such as underdamped Langevin (kinetic Langevin), for which analogous mutual information guarantees are not provided here.

Understanding I(X0; Xt) along these alternative chains would extend the scope of independence-time guarantees beyond the settings covered in the paper.

References

While we study the convergence of mutual information along the Langevin diffusion and ULA, many other interesting questions remain open. Second, we can study the convergence of mutual information along other Markov chains, such as the underdamped Langevin dynamics.

— Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of $Φ$-Mutual Information (2402.17067 - Liang et al., 26 Feb 2024) in Discussion (Section: Discussion)