Properties of Quick Simulation Random Fields

TL;DR

This paper introduces a new class of discrete random fields that enable quick, single-pass simulation and covariance inference, improving efficiency over traditional multi-pass methods like MCMC, with applications in data authentication and image generation.

Contribution

It presents a novel construction method for correlated random fields that simplifies simulation while maintaining specified marginals and covariances, including a condition for Markov random fields.

Findings

Single-pass simulation algorithm developed

Conditions for covariance and marginal probability compatibility analyzed

Markov random fields identified as a subclass with a natural condition

Abstract

Herein, we introduce and study a new class of discrete random fields designed for quick simulation and covariance inference under inhomogeneous condition. Simulation of these correlated fields can be done in a single pass instead of relying on multi-pass convergent methods like the Gibbs Sampler or other Markov Chain Monte Carlo methods. The fields are constructed directly from specified marginal probability mass functions and covariances between nearby sites. The proposition on which the construction is based establishes when and how it is possible to simplify the conditional probabilities of each site given the other sites in a manner that makes simulation quite feasible yet maintains desired marginal probabilities and covariances between sites. Special cases of these correlated fields have been deployed successfully in data authentication, object detection and image generation. The limitations that must be imposed on the covariances and marginal probabilities in order for the algorithm to work are studied. What's more, a necessary and sufficient condition that guarantees the permutation property of correlated random fields are investigated. In particular, Markov random fields as a subclass of correlated random fields are derived by a general and natural condition. Consequently, a direct and flexible single pass algorithm for simulating Markov random fields follows.