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Perfect Sampling for Hard Spheres from Strong Spatial Mixing

Konrad Anand, Andreas Göbel, Marcus Pappik, Will Perkins

TL;DR

This work proves that strong spatial mixing implies the existence of a perfect sampling algorithm for the hard-sphere Gibbs point process with linear expected running time in the region’s volume. The method discretizes space into boxes and uses Bayes filters to perform local, bias-corrected updates, implemented via Bernoulli factories to avoid exact knowledge of update probabilities. The approach extends to Gibbs point processes with finite-range, repulsive potentials, yielding near-linear performance when the potential satisfies $(a,b)$-SSM up to the activity $\lambda$; explicit bounds are provided in terms of the temperedness constant $C_{\phi}$ and related constants. Practically, this delivers provably exact samples in parameter regimes where fast approximate samplers exist, with performance guarantees that scale linearly with volume, up to polylog factors in the most general setting. The integration of Bernoulli factories into a continuum perfect-sampling framework represents a significant methodological advance for spatial point processes.

Abstract

We provide a perfect sampling algorithm for the hard-sphere model on subsets of $\mathbb{R}^d$ with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.

Perfect Sampling for Hard Spheres from Strong Spatial Mixing

TL;DR

This work proves that strong spatial mixing implies the existence of a perfect sampling algorithm for the hard-sphere Gibbs point process with linear expected running time in the region’s volume. The method discretizes space into boxes and uses Bayes filters to perform local, bias-corrected updates, implemented via Bernoulli factories to avoid exact knowledge of update probabilities. The approach extends to Gibbs point processes with finite-range, repulsive potentials, yielding near-linear performance when the potential satisfies -SSM up to the activity ; explicit bounds are provided in terms of the temperedness constant and related constants. Practically, this delivers provably exact samples in parameter regimes where fast approximate samplers exist, with performance guarantees that scale linearly with volume, up to polylog factors in the most general setting. The integration of Bernoulli factories into a continuum perfect-sampling framework represents a significant methodological advance for spatial point processes.

Abstract

We provide a perfect sampling algorithm for the hard-sphere model on subsets of with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.
Paper Structure (26 sections, 47 theorems, 218 equations, 2 figures, 5 algorithms)

This paper contains 26 sections, 47 theorems, 218 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. The hard-sphere model, strong spatial mixing, and perfect sampling
  3. Gibbs point processes with finite-range repulsive potentials
  4. Related work and future directions
  5. Outline of the paper
  6. Intuitive idea behind the algorithm
  7. Bayes filter for the hard-sphere model
  8. Bayes filter for general potentials
  9. Preliminaries
  10. Gibbs point processes
  11. Bernoulli factories
  12. The algorithm
  13. Proof of correctness
  14. Strong spatial mixing and success probabilities of Bayes filters
  15. Strong spatial mixing and point density functions
  16. ...and 11 more sections

Key Result

Theorem 1

There is a perfect sampling algorithm for the hard-sphere model on finite boxes $\Lambda\xspace \subset \mathds{R}\xspace^d$ with the property that if the hard-sphere model exhibits $(a,b)$-strong spatial mixing up to $\lambda$, then the expected running time of the algorithm at activity $\lambd

Figures (2)

  • Figure 1: Example for the regions considered throughout the proof of \ref{['lemma:invariant_case1']} for $\ell\xspace = 2$.
  • Figure 2: Example for the regions considered throughout the proof of \ref{['lemma:invariant_case0']} for $\ell\xspace = 2$.

Theorems & Definitions (96)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Remark 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • proof
  • ...and 86 more