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This paper proves a universality theorem for conditional diffusion models where the reverse kernels are finite Gaussian mixtures with ReLU-network logits. It shows that such models can approximate regular target distributions well in context-averaged conditional KL divergence, up to a terminal mismatch error. The proof uses a path-space decomposition to reduce the problem to per-step reverse-kernel errors, which are then bounded using Gaussian-mixture theory and quantitative ReLU bounds.
ReLU-based Gaussian mixture reverse kernels in conditional diffusion models can provably approximate any suitably regular target distribution, offering a theoretical foundation for their widespread use.
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.