Score Change of Variables

Abstract

We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation y = (x), the transformed score function _y q(y) can be expressed directly in terms of _x p(x). Using this result, we develop two applications: First, we establish a reverse-time It\^o lemma for score-based diffusion models, allowing the use of _x p_t(x) to reverse an SDE in the transformed space without directly learning _y q_t(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.

Tasks

Reproductions