证明了条件扩散模型用高斯混合反向核可逼近任意光滑目标分布。
Universality of Gaussian-Mixture Reverse Kernels in Conditional Diffusion
- 用带ReLU网络的有限高斯混合反向核建模扩散过程
- 在增加扩散时长时，逼近误差趋近于零
- 适合研究扩散模型理论性质的学者
我们证明，若条件扩散模型的反向核为具有ReLU网络对数似然的有限高斯混合，则其在上下文平均条件KL散度下可任意逼近合适正则的目标分布，仅受不可约终端不匹配限制；该不匹配通常随扩散时长增加而消失。路径空间分解将输出误差归结为该不匹配与每步反向核误差之和；假设每步反向核通过有限维特征映射，问题转化为静态条件密度逼近，结合Norets高斯混合理论与量化ReLU界求解。在精确终端匹配下，所得神经反向核类在条件KL意义下稠密。
原文摘要 · Abstract (English)
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.
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