用因果最优传输解决奇异扩散模型的条件生成难题
Conditioning Degenerate Diffusion Models
- 引入因果最优传输构建近似损失函数
- 在密度不存在或不光滑时仍能实现最小熵控制
- 适用于低平滑性条件分布，适合理论研究者
当前条件生成模型高度依赖训练过程中的得分函数进行引导。当生成模型为扩散过程且扩散系数奇异，且底层（条件）分布不存在或不光滑时，我们使用因果最优传输定义近似损失函数，识别出在最弱假设下的最小熵控制。该方法基于因果最优传输及其通过预测表示性质表征的（条件）扩散过程，其相关鞅问题如Ustunel所述是适定的。
原文摘要 · Abstract (English)
Current conditioned generative models heavily rely on score functions for guidance during training. When the generative model is a diffusion process with a singular diffusion coefficient and the underlying (conditional) densities either do not exist or are not smooth, we use causal optimal transport to define \emph{approximate} loss functions that identify a minimum-entropy control for guidance under minimal assumptions. Our approach relies on causal optimal transport and its characterization through the predictable representation property of (conditioned) diffusion processes whose associated martingale problem is well posed, \`a la \"Ust\"unel.
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