Universality of Gaussian-Mixture Reverse Kernels in Conditional Diffusion
Abstract
We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up to an irreducible terminal mismatch that typically vanishes with increasing diffusion horizon. A path-space decomposition reduces the output error to this mismatch plus per-step reverse-kernel errors; assuming each reverse kernel factors through a finite-dimensional feature map, each step becomes a static conditional density approximation problem, solved by composing Norets' Gaussian-mixture theory with quantitative ReLU bounds. Under exact terminal matching the resulting neural reverse-kernel class is dense in conditional KL.
Keywords
Cite
@article{arxiv.2604.13470,
title = {Universality of Gaussian-Mixture Reverse Kernels in Conditional Diffusion},
author = {Nafiz Ishtiaque and Syed Arefinul Haque and Kazi Ashraful Alam and Fatima Jahara},
journal= {arXiv preprint arXiv:2604.13470},
year = {2026}
}
Comments
10+19 pages