[Paper Review] Perfect sampling from spatial mixing

Weiming Feng, Heng Guo+1 more|arXiv (Cornell University)|Jul 13, 2019

Markov Chains and Monte Carlo Methods39 references4 citations

TL;DR

This paper introduces a novel perfect sampling algorithm for Gibbs distributions in spin systems that runs in linear expected time when correlation decays faster than neighborhood growth. It leverages conditional independence and spatial mixing to achieve efficient, interruptible sampling, with optimal performance on lattice graphs like $\mathbb{Z}^d$ under strong spatial mixing.

ABSTRACT

We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with sub-exponential neighborhood growth like $\mathbb{Z}^d$, our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer-dimer models in such graphs.

Motivation & Objective

• To develop a perfect sampling algorithm for general Gibbs distributions that is efficient and interruptible.
• To establish a connection between spatial mixing properties and linear-time perfect sampling.
• To extend efficient perfect sampling to dynamic settings where the Gibbs distribution changes over time.
• To achieve optimal runtime for well-known models like colorings and monomer-dimer systems on $\mathbb{Z}^d$.
• To generalize the approach to multi-body interactions via hypergraph spin systems.

Proposed method

• The algorithm uses a block-based update strategy based on local neighborhoods, where each update is performed on a ball of radius $\ell$ around a randomly selected vertex.
• It maintains a set $\mathcal{R}$ of vertices that may need re-sampling, initialized as $\mathcal{D} \cup \partial\mathcal{D}$ after a system update.
• At each step, a vertex $u \in \mathcal{R}$ is chosen uniformly at random, and the configuration on its $\ell$-neighborhood is resampled conditionally on the current boundary values.
• The resampling step uses rejection sampling with a lower bound on the conditional probability to ensure correctness and maintain the target distribution.
• The algorithm ensures correctness via the conditional independence property of Gibbs distributions and guarantees exact samples upon termination.
• It is interruptible: any intermediate termination yields a valid sample from the target distribution.

Experimental results

Research questions

• RQ1 Can perfect sampling for general Gibbs distributions be achieved in linear time under spatial mixing?
• RQ2 Does spatial mixing imply efficient perfect sampling when the correlation decay rate exceeds neighborhood growth?
• RQ3 Can the algorithm be made interruptible and efficiently adapted to dynamic changes in the system?
• RQ4 Can the method be extended to systems with multi-body interactions?
• RQ5 What is the runtime complexity of perfect sampling for colorings and monomer-dimer models on $\mathbb{Z}^d$ under spatial mixing?

Key findings

• The algorithm runs in $O(n)$ expected time for graphs with sub-exponential neighborhood growth, such as $\mathbb{Z}^d$, when strong spatial mixing holds with a rate faster than neighborhood growth.
• For lattice graphs like $\mathbb{Z}^d$, the algorithm achieves linear time if the Gibbs sampler has $O(n\log n)$ mixing time.
• The algorithm is interruptible: any early termination still yields a correct sample from the target distribution.
• It supports dynamic updates to the system, maintaining perfect sampling efficiency under spatial mixing in the updated model.
• The method generalizes to hypergraph spin systems with multi-body interactions under an appropriate variant of spatial mixing.
• The approach yields the currently best perfect samplers for list colorings and monomer-dimer models on $\mathbb{Z}^d$.