Understanding Diffusion Models by Feynman's Path Integral

TL;DR

This paper introduces a novel Feynman path integral formulation for diffusion models, providing insights into the performance differences between stochastic and deterministic sampling, and applies quantum physics techniques to evaluate model likelihoods.

Contribution

The paper develops a Feynman path integral framework for diffusion models, linking stochastic and deterministic schemes, and uses quantum physics methods to analyze their performance.

Findings

Path integral formulation describes diffusion models comprehensively.

Interpolating parameter acts as a Planck's constant analogue.

WKB expansion evaluates negative log-likelihood for performance analysis.

Abstract

Score-based diffusion models have proven effective in image generation and have gained widespread usage; however, the underlying factors contributing to the performance disparity between stochastic and deterministic (i.e., the probability flow ODEs) sampling schemes remain unclear. We introduce a novel formulation of diffusion models using Feynman's path integral, which is a formulation originally developed for quantum physics. We find this formulation providing comprehensive descriptions of score-based generative models, and demonstrate the derivation of backward stochastic differential equations and loss functions.The formulation accommodates an interpolating parameter connecting stochastic and deterministic sampling schemes, and we identify this parameter as a counterpart of Planck's constant in quantum physics. This analogy enables us to apply the Wentzel-Kramers-Brillouin (WKB) expansion, a well-established technique in quantum physics, for evaluating the negative log-likelihood to assess the performance disparity between stochastic and deterministic sampling schemes.