Reflected diffusion study reveals hidden boundary errors masked by hard reflection

In a new arXiv paper, Ziyue Wang and Takafumi Kanamori investigate a subtle but critical issue in reflected diffusion models: whether the learned score correctly matches the boundary behavior of the forward process. Reflected diffusion keeps samples within bounded domains, but standard implicit score matching leaves an integration-by-parts boundary term. The authors prove this term reduces to a single scalar per boundary point—the diffusion-weighted normal component of the score, called the conormal trace. Under anisotropic diffusion, this trace generally differs from the ordinary normal score component, and the no-flux condition fixes its value while leaving other boundary components free.

The paper's key theoretical contribution is a parametrization that exactly enforces the required conormal trace on hyperrectangles without adding trainable parameters or stochastic boundary estimators. Under regularity assumptions, this construction can represent the true score, whereas fixing an incorrect trace creates a bias that more training data cannot remove. The authors extend the approach to simplices and polygonal domains, and introduce "reflection masking"—showing that hard reflection keeps samples feasible even when the learned trace is wrong. Experiments with less frequent reflection, anisotropic diffusion, and mass near constraint intersections show the clearest separation; under full reflection, final sample placement improves inconsistently, demonstrating that hard repair can hide boundary-score errors and decouple score accuracy from generation quality.