# Bounds on Lifting Continuous-State Markov Chains to Speed Up Mixing

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## Abstract

It is often possible to speed up the mixing of a Markov chain \(\{ X_{t} \}_{t \in \mathbb {N}}\) on a state space \(\Omega \) by *lifting*, that is, running a more efficient Markov chain \(\{ \widehat{X}_{t} \}_{t \in \mathbb {N}}\) on a larger state space \(\hat{\Omega } \supset \Omega \) that projects to \(\{ X_{t} \}_{t \in \mathbb {N}}\) in a certain sense. Chen et al. (Proceedings of the 31st annual ACM symposium on theory of computing. ACM, 1999) prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure of \(\{X_t\}_{t \in \mathbb {N}}\). Unfortunately, this extra factor makes the bound in Chen et al.
(1999) very loose for Markov chains on large state spaces and useless for Markov chains on continuous state spaces. In this paper, we develop an extension of the evolving set method that allows us to refine this extra factor and find bounds for Markov chains on continuous state spaces that are analogous to the bounds in Chen et al.
(1999). These bounds also allow us to improve on the bounds in Chen et al.
(1999) for some chains on finite state spaces.

## Keywords

Markov chain Nonreversible Lift Mixing time Conductance## Mathematics Subject Classification (2010)

60J05## Notes

### Acknowledgements

The first author, KR, is partially supported by NSF Grant DMS-1407504 and the second author, AMS, is partially supported by an NSERC Grant. This project was started while AMS was visiting ICERM, and he thanks ICERM for its generous support.

## References

- 1.Chen, F., Lovász, L., Pak, I.: Lifting markov chains to speed up mixing. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 275–281. ACM (1999)Google Scholar
- 2.Diaconis, P.: Group Representations in Probability and Statistics, volume 11 of IMS Lecture Notes—Monograph Series. Institute of Mathematical Statistics, Hayward (1988)Google Scholar
- 3.Diaconis, P., Holmes, S., Neal, R.M.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab.
**10**(3), 726–752 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc.
**46**(1), 179–205 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Diaconis, P., Miclo, L.: On the spectral analysis of second-order Markov chains. Ann. Fac. Sci. Toulouse Math.
**22**, 573–621 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev.
**70**(3), 419–435 (2002)CrossRefzbMATHGoogle Scholar - 7.Gomez, F., Schmidhuber, J., Sun, Y.: Improving the asymptotic performance of Markov chain Monte-Carlo by inserting vortices. Advances in Neural Information Processing Systems (2010)Google Scholar
- 8.Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
- 9.Morris, B., Peres, Y.: Evolving sets, mixing and heat kernel bounds. Probab. Theory Relat. Fields
**133**, 245–266 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Neal, R.: Improving asymptotic variance of MCMC estimators: Non-reversible chains are better. Technical Report No. 0406, Department of Statistics, University of Toronto (2004)Google Scholar
- 11.Peres, Y., Sousi, P.: Mixing times are hitting times of large sets. J. Theor. Probab.
**28**(2), 488–519 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Rosenthal, J.: Random rotations: characters and random walks on \(\text{ SO }(n)\). Ann. Probab.
**22**, 398–423 (1994)MathSciNetCrossRefzbMATHGoogle Scholar