Path convergence in diffusion models

Roi Holtzman; Roman Beauvallet; Werner Krauth

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arXiv · 2606.12161

Path convergence in diffusion models

Roi Holtzman·Roman Beauvallet·Werner Krauth

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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/sqrt(p), but with infinite mean square deviation. We demonstrate that the path convergence allows for extrapolation towards the p=infinity 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.

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Roi Holtzman, Roman Beauvallet, Werner Krauth. 2026-06-10. Path convergence in diffusion models. https://arxiv.org/abs/2606.12161

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