Universality of Gaussian-Mixture Reverse Kernels in Conditional Diffusion

arXiv cs.LG / 4/16/2026

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Key Points

The paper proves that conditional diffusion models with reverse kernels implemented as finite Gaussian mixtures (with ReLU-network logits) can approximate regular target distributions arbitrarily well in context-averaged conditional KL divergence.
Error is decomposed into an irreducible terminal mismatch term (often shrinking as the diffusion horizon increases) plus per-step reverse-kernel approximation errors.
By using path-space decomposition and assuming each reverse kernel factors through a finite-dimensional feature map, the authors reduce each diffusion step to a static conditional density approximation problem.
The approach combines Norets’ Gaussian-mixture approximation framework with quantitative ReLU bounds to control the per-step errors, and under exact terminal matching shows the neural reverse-kernel class is dense in conditional KL.
The results provide a universality/density guarantee that strengthens theoretical foundations for representational capacity in conditional diffusion with Gaussian-mixture reverse transitions.

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.