Score Change of Variables
This work addresses foundational challenges in machine learning for researchers and practitioners in score-based models and density estimation, offering incremental theoretical extensions with practical implications.
The paper tackles the problem of deriving a change of variables formula for score functions, enabling applications such as decoupling training and sampling in diffusion models and extending sliced score matching to arbitrary transformations, with empirical comparisons showing improved flexibility in high-dimensional density estimation.
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.