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We extend results of Richard Holley beyond the integer lattice to a large class of countable groups which includes free groups and all amenable groups: for nearest-neighbor interactions on the Cayley graphs of such groups, we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-invariant measure converges weakly to the set of Gibbs measures when evolved under the corresponding Glauber dynamics. These results are proven using a notion of free energy density relative to a sofic approximation by homomorphisms, which avoids the boundary problems which appear when applying a standard free energy method in a nonamenable setting. We also show that any measure which minimizes this free energy density is Gibbs.
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Second order ensemble Langevin method for sampling and inverse problems
We propose a sampling method based on an ensemble approximation of second order Langevin dynamics. The log target density is appended with a quadratic term in an auxiliary momentum variable and damped-driven Hamiltonian dynamics introduced; the resulting stochastic differential equation is invariant to the Gibbs measure, with marginal on the position coordinates given by the target. A preconditioner based on covariance under the law of position coordinates under the dynamics does not change this invariance property, and is introduced to accelerate convergence to the Gibbs measure. The resulting mean-field dynamics may be approximated by an ensemble method; this results in a gradient-free and affine-invariant stochastic dynamical system with desirable provably uniform convergence properties across the class of all Gaussian targets. Numerical results demonstrate the potential of the method as the basis for a numerical sampler in Bayesian inverse problems, beyond the Gaussian setting.
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- Award ID(s):
- 1835860
- PAR ID:
- 10637593
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Communications in Mathematical Sciences
- Volume:
- 23
- Issue:
- 5
- ISSN:
- 1539-6746
- Page Range / eLocation ID:
- 1299-1317
- Subject(s) / Keyword(s):
- Bayesian inverse problems sampling ensemble method second order Langevin equation Hybrid Monte Carlo mean-field models nonlinear Fokker-Planck equation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/CMS.250517001811
