Score Change of Variables

We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation y = φ(x), the transformed score function nablay log q(y) can be expressed directly in terms of nablax log p(x). Using this result, we develop two applications: First, we establish a reverse-time Ito lemma for score-based diffusion models, allowing the use of nablax log pt(x) to reverse an SDE in the transformed space without directly learning nablay log qt(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.