Generating from Discrete Distributions Using Diffusions: Insights from Random Constraint Satisfaction Problems

TL;DR

This paper investigates how diffusion-based generative methods perform on random constraint satisfaction problems like k-SAT, revealing insights that challenge intuition and improve generation accuracy.

Contribution

It connects random CSP theory with diffusion techniques, showing how insights can enhance discrete data generation and outperform existing methods.

Findings

Continuous diffusions outperform discrete diffusions

Learned diffusions can match theoretical accuracy

Smart variable ordering improves generation accuracy

Abstract

Generating data from discrete distributions is important for a number of application domains including text, tabular data, and genomic data. Several groups have recently used random $k$-satisfiability ($k$-SAT) as a synthetic benchmark for new generative techniques. In this paper, we show that fundamental insights from the theory of random constraint satisfaction problems have observable implications (sometime contradicting intuition) on the behavior of generative techniques on such benchmarks. More precisely, we study the problem of generating a uniformly random solution of a given (random) $k$-SAT or $k$-XORSAT formula. Among other findings, we observe that: $(i)$~Continuous diffusions outperform masked discrete diffusions; $(ii)$~Learned diffusions can match the theoretical `ideal' accuracy; $(iii)$~Smart ordering of the variables can significantly improve accuracy, although not following popular heuristics.