Diffusion Models Break the Curse of Dimensionality for Mixed-Smooth Besov Densities

Diffusion models learn a probability law by adding noise, learning how to denoise, and then running the noising process backward. In high dimensions, classical nonparametric theory predicts a curse: the sample-size exponent worsens with the ambient dimension. We show that this curse can be provably broken when the density has mixed smoothness, meaning that simultaneous high-frequency variation across many coordinates is strongly limited, the structure exploited by sparse grids. On the torus, for densities in a mixed Besov ball, we prove a total-variation minimax rate with the one-dimensional polynomial exponent $n^{-s/(2s+1)}$ and only logarithmic dependence on the dimension. We then show that a diffusion estimator using a sparse-grid score approximator attains this rate, up to standard algorithmic polylogarithms, under explicit score regularity and multiplier assumptions; for $s>1/2$ the multiplier condition holds on the whole class, so the rate is unconditional there. A controlled study reproduces the predicted scaling. Our analysis also reveals a new Wasserstein geometry: transport is governed by the finest coordinate scale, leaving sharp diffusion $W_1$ rates an important open direction.