Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sampling on Discrete Spaces with Temporal Point Processes

Cameron A. Stewart, Maneesh Sahani

TL;DR

This work shows how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity.

Abstract

Temporal point processes offer a powerful framework for sampling from discrete distributions, yet they remain underutilized in existing literature. We show how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. Structured as a system of potentially coupled infinite-server queues with deterministic service times, the sampler exhibits a discrete form of momentum that suppresses random-walk behaviour. The admissible families of processes permit both reversible and non-reversible dynamics. As an application, we derive a recurrent stochastic neural network whose dynamics implement sampling-based computation and exhibit some biologically plausible features, including relative refractory periods and oscillatory dynamics. The introduction of auxiliary randomness reduces the sampler to a birth-death process, establishing the latter as a degenerate case with the same limiting distribution. In simulations on 63 target distributions, our sampler always outperforms these birth-death processes and frequently outperforms Zanella processes in multivariate effective sample size, with further gains when normalized by CPU time.

Sampling on Discrete Spaces with Temporal Point Processes

TL;DR

This work shows how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity.

Abstract

Temporal point processes offer a powerful framework for sampling from discrete distributions, yet they remain underutilized in existing literature. We show how to construct, for any target multivariate count distribution with downward-closed support, a multivariate temporal point process whose event-count vector in a fixed-length sliding window converges in distribution to the target as time tends to infinity. Structured as a system of potentially coupled infinite-server queues with deterministic service times, the sampler exhibits a discrete form of momentum that suppresses random-walk behaviour. The admissible families of processes permit both reversible and non-reversible dynamics. As an application, we derive a recurrent stochastic neural network whose dynamics implement sampling-based computation and exhibit some biologically plausible features, including relative refractory periods and oscillatory dynamics. The introduction of auxiliary randomness reduces the sampler to a birth-death process, establishing the latter as a degenerate case with the same limiting distribution. In simulations on 63 target distributions, our sampler always outperforms these birth-death processes and frequently outperforms Zanella processes in multivariate effective sample size, with further gains when normalized by CPU time.
Paper Structure (19 sections, 8 theorems, 47 equations, 1 figure)

This paper contains 19 sections, 8 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Point processes for sampling
  4. Properties of the point process sampler
  5. Time reversal
  6. Connection to sampling with birth-death processes
  7. Application: a stochastic neural network model
  8. Sampling efficiency simulation study
  9. Study design
  10. Poisson distribution
  11. Sherrington-Kirkpatrick model
  12. Stochastic neural network model
  13. Commonly observed behaviour
  14. Discussion
  15. Proofs
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f$ define $\pi$ according to eqn:mass. If for $t \in \mathbb{R}_{\geq m}$, then $\pi$ is the limiting distribution of $S(t)$ as $t\uparrow\infty$.

Figures (1)

  • Figure 1: Experimentally derived efficiency estimates for five different jump process samplers applied to 63 different target distributions. Error bars show 95% confidence intervals, but in most cases they are smaller than the markers and hence not visible. Left column: Poisson distribution (log-log plots). Middle column: Sherrington-Kirkpatrick model (log-linear plots). Right column: stochastic neural network model (log-linear plots). Top row: ess for $1\,000$ sequential weighted samples. Bottom row: ess per CPU second.

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Lemma 1
  • Example 1: The Poisson Distribution and Poisson Processes
  • Example 2: Multivariate Binary Support and Boltzmann Machines
  • Lemma 2
  • proof
  • Proposition 1
  • Proposition 2
  • Lemma 1: Restated
  • ...and 5 more