推导得分函数变量变换通用公式，实现生成模型跨空间采样与更灵活的密度估计。
Score Change of Variables
- 建立得分函数在光滑可逆变换下的直接转换关系。
- 支持在不同空间训练与采样，解耦扩散模型的正向与反向过程。
- 扩展切片得分匹配至任意光滑变换，提升高维密度估计灵活性。
我们推导了得分函数的一般变量变换公式，证明对于光滑可逆变换 $\mathbf{y} = ϕ(\mathbf{x})$，变换后的得分函数 $\nabla_{\mathbf{y}} \log q(\mathbf{y})$ 可直接由 $\nabla_{\mathbf{x}} \log p(\mathbf{x})$ 表示。基于此，我们发展两项应用：第一，建立得分型扩散模型的逆时间伊藤引理，使得可在变换空间中利用 $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ 反演 SDE，无需直接学习 $\nabla_{\mathbf{y}} \log q_t(\mathbf{y})$，从而实现训练与采样空间解耦；第二，提出广义切片得分匹配，将传统线性投影扩展至任意光滑变换，增强高维密度估计的表达能力。我们在概率单纯形上的扩散模型中验证理论进展，并与传统切片得分匹配方法进行实证比较。
原文摘要 · Abstract (English)
We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation $\mathbf{y} = ϕ(\mathbf{x})$, the transformed score function $\nabla_{\mathbf{y}} \log q(\mathbf{y})$ can be expressed directly in terms of $\nabla_{\mathbf{x}} \log p(\mathbf{x})$. Using this result, we develop two applications: First, we establish a reverse-time Itô lemma for score-based diffusion models, allowing the use of $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ to reverse an SDE in the transformed space without directly learning $\nabla_{\mathbf{y}} \log q_t(\mathbf{y})$. This approach enables training diffusion models in one space but sampling in another, effectively decoupling the forward and reverse processes. Second, we introduce generalized sliced score matching, extending traditional sliced score matching from linear projections to arbitrary smooth transformations. This provides greater flexibility in high-dimensional density estimation. We demonstrate these theoretical advances through applications to diffusion on the probability simplex and empirically compare our generalized score matching approach against traditional sliced score matching methods.
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