Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler
Researchers prove local geometric mixing via Dobrushin contraction for Diffusion Path Monte Carlo and the Proximal Sampler.
The paper defines local geometric mixing as geometric convergence in total variation over finitely many transitions, capturing fast local equilibration when global mixing is slower. Bounds are derived using Dobrushin contraction. The authors apply the framework to Diffusion Path Monte Carlo, whose ideal transitions match the Proximal Sampler, covering both the ideal chain and a Metropolis-adjusted implementation. For the ideal method, the guarantees complement recent spectral-gap estimates and yield mixing-time bounds.
- Local geometric mixing requires fast convergence over only finitely many transitions.
- Dobrushin contraction yields local mixing bounds under minimal assumptions.
- Analysis covers ideal Diffusion Path Monte Carlo and its Metropolis-adjusted version.
- Ideal-method guarantees complement spectral-gap estimates with mixing-time bounds.
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Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.28338