Cross-fluctuation phase transitions reveal sampling dynamics in diffusion models

TL;DR

This paper introduces a novel analysis of diffusion model sampling dynamics using cross-fluctuations, revealing discrete phase transitions that improve efficiency and enable zero-shot tasks.

Contribution

It uncovers phase transition phenomena in diffusion models through cross-fluctuations, providing a new perspective that enhances sampling and task performance.

Findings

Detects sharp transitions as discontinuities in cross-fluctuations.

Derives a closed-form for cross-fluctuations in variance-preserving SDEs.

Improves sampling efficiency and zero-shot task performance without retraining.

Abstract

We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using cross-fluctuations, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic normal distribution, samples undergo sharp, discrete transitions, eventually forming distinct events of a desired distribution while progressively revealing finer structure. As this process is reversible, these transitions also occur in reverse, where intermediate states progressively merge, tracing a path back to the initial distribution. We demonstrate that these transitions can be detected as discontinuities in $n^{\text{th}}$-order cross-fluctuations. For variance-preserving SDEs, we derive a closed-form for these cross-fluctuations that is efficiently computable for the reverse trajectory. We find that detecting these transitions directly boosts sampling efficiency, accelerates class-conditional and rare-class generation, and improves two zero-shot tasks--image classification and style transfer--without expensive grid search or retraining. We also show that this viewpoint unifies classical coupling and mixing from finite Markov chains with continuous dynamics while extending to stochastic SDEs and non Markovian samplers. Our framework therefore bridges discrete Markov chain theory, phase analysis, and modern generative modeling.