Path convergence in diffusion models

Abstract

Abstract We discuss diffusion-model paths interpolating between a target distribution known only through p patterns and a reference distribution that can be sampled. These interpolating paths can be constructed symmetrically or else in forward direction (often referred to as a "noising") from the target patterns to the reference distribution or in backward direction (as a "denoising") from the reference distribution to the patterns. For backward paths with identical diffusion noise, we consider the path convergence in number of patterns p towards the path for infinitely many patterns. In a one-dimensional test case, we show that this convergence is on a scale 1/ √ p, but with infinite mean square deviation on that scale. We demonstrate that the path convergence allows for extrapolation towards the p = ∞ path which samples the target distribution. We provide a proof-of-concept extrapolation algorithm and propose the convergence and extrapolation of paths as a possible strategy for density estimation and generalization. We illustrate all our algorithms through pseudo-codes and provide Python implementations.