Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions

arXiv:2402.15602v1 Announce Type: cross
Abstract: We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and $d$ represent the sample size and the dimension, $t$ is bounded above and below by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As a consequence, this yields …

abstract arxiv assumptions beyond cs.lg diffusion diffusion model diffusion models error function kernel math.st mean minimax non-parametric parametric perspective sample sampling show square statistics stat.ml stat.th study type