Diffusion models generate data by learning the score of a forward noising process, but the sample complexity of this score estimation step is not fully understood. In a new preprint, Fan Liu, Yahong Yang, and Wei Zhu analyze denoising score matching (DSM) with finite-width shallow ReLU networks for the Ornstein–Uhlenbeck process in ℝᵈ. The authors derive explicit Sobolev bounds on the score estimation error that are uniform in time on each fixed ball, combining estimates from the Gaussian transition kernel and the initial density.
Under weak differentiability and polynomial-growth conditions on the initial density, the paper bounds approximation error using existing shallow-network theory and gives a sufficient path-norm budget. For empirical minimizers with unbounded inputs and labels, the authors use truncation and localized Rademacher complexity to obtain high-probability risk bounds. With additional bounded-density and sub-Gaussian assumptions, the squared L²(P_t) score risk is shown to be Õ(n^{−(d+3)/(2d+3)}), where n is the number of training samples.
By aggregating these score errors over a time grid and applying existing sampling theory with early stopping, the authors obtain a high-probability total-variation error of Õ(n^{−(d+3)/(4d+6)}) between the generated and original data distributions. The nonasymptotic bounds retain explicit dependence on diffusion time and on the regularity and tail parameters of the initial distribution. The paper thus provides a complete finite-sample analysis for shallow-network score estimation, clarifying how width, path norm, and sampling parameters trade off against generation quality.