Path convergence in diffusion models
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.
Versions
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BibTeX
@misc{holtzman2026path,
title = {{Path convergence in diffusion models}},
author = {Roi Holtzman and Roman Beauvallet and Werner Krauth},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.12161},
eprint = {2606.12161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12161},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Path convergence in diffusion models
AU - Roi Holtzman
AU - Roman Beauvallet
AU - Werner Krauth
PY - 2026
DA - 2026/06/10
PB - arXiv
DO - 10.48550/arxiv.2606.12161
UR - https://arxiv.org/abs/2606.12161
AB - 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.
ER -