Mixing and concentration by Ricci curvature

Paulin, Daniel

doi:10.1016/j.jfa.2015.12.010

Cited by 18 publications

(17 citation statements)

References 44 publications

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“…We apply it to hypothesis testing for coin tossing (Example 3.24). Another application is given in Paulin (2013), where we estimate the pseudo spectral gap of the Glauber dynamics with systemic scan in the case of the Curie-Weiss model. In these examples, the spectral gap of the multiplicative reversiblization is 0, but the pseudo spectral gap is positive.…”

Section: Preliminariesmentioning

confidence: 99%

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Paulin

2015

Electron. J. Probab.

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We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the "pseudo spectral gap", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains.Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.

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Section: Preliminariesmentioning

confidence: 99%

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Paulin

2015

Electron. J. Probab.

Self Cite

172

193

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“…In [29], Joulin and Ollivier have shown that strict Kantorovich contractivity of the transition kernel implies bounds for the variance and concentration estimates for ergodic averages of a Markov chain. Their results have since been extended to cover more general frameworks by Paulin [38]. More recently, Pillai and Smith [39] as well as Rudolf and Schweizer [40] have developed a perturbation theory for Markov chains that are contractive in a Kantorovich distance, cf.…”

Section: Introductionmentioning

confidence: 99%

Quantitative contraction rates for Markov chains on general state spaces

Eberle¹,

Majka²

2019

Electron. J. Probab.

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We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich (L 1 Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on R d with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.2010 Mathematics Subject Classification. 60J05, 60J22, 65C05, 65C30, 65C40.

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“…Consequently we have that for u ≥ t = (1 + )(1/k)n log n u satisfies (20) for some sufficiently large c > c Γ . Hence lim sup n→∞dT V (t ) → 0 and thus (17) holds, which shows Theorem 1.1 for conjugacy classes such that the limit in (C) exists and |Γ| = o(n).…”

Section: Curvature Implies Mixingmentioning

confidence: 95%

Cutoff for conjugacy-invariant random walks on the permutation group

Berestycki

Sengul

2018

Probab. Theory Relat. Fields

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We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm as well as contractivity of the distance. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time cn/2, the curvature is asymptotically zero for c ≤ 1 and is strictly positive for c > 1.

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Mixing and concentration by Ricci curvature

Cited by 18 publications

References 44 publications

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Concentration inequalities for Markov chains by Marton couplings and spectral methods

Quantitative contraction rates for Markov chains on general state spaces

Cutoff for conjugacy-invariant random walks on the permutation group

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