Non-reversible lifts of reversible diffusion processes and relaxation times
Andreas Eberle, Francis Lörler
TL;DR
The paper develops a formal framework for second-order lifts of reversible diffusions, showing that many non-reversible processes used in applications arise as lifts of simple reversible dynamics. It introduces non-asymptotic relaxation times and proves a square-root speed-up bound for lifts, relating lift performance to the singular-value gap of the generator. By embedding space-time Poincaré inequalities into the lift language, it derives explicit upper bounds on relaxation times and identifies optimal lift structures, notably showing near-optimality of Randomised Hamiltonian Monte Carlo in Gaussian/convex settings. The results unify hypocoercivity techniques with lift theory, providing practical criteria to design and analyze efficient non-reversible samplers and revealing how to choose refresh rates to maximize convergence speed. Overall, the work offers a principled pathway to understand, quantify, and optimize non-reversible acceleration via lifts of simple reversible diffusions.
Abstract
We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincaré inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.