Few-Shot Diffusion Models Escape the Curse of Dimensionality

While diffusion models have demonstrated impressive performance, there is a growing need for generating samples tailored to specific user-defined concepts. The customized requirements promote the development of few-shot diffusion models, which use limited $n_{ta}$ target samples to fine-tune a pre-trained diffusion model trained on $n_s$ source samples. Despite the empirical success, no theoretical work specifically analyzes few-shot diffusion models. Moreover, the existing results for diffusion models without a fine-tuning phase can not explain why few-shot models generate great samples due to the curse of dimensionality. In this work, we analyze few-shot diffusion models under a linear structure distribution with a latent dimension $d$. From the approximation perspective, we prove that few-shot models have a $\widetilde{O}(n_s^{-2/d}+n_{ta}^{-1/2})$ bound to approximate the target score function, which is better than $n_{ta}^{-2/d}$ results. From the optimization perspective, we consider a latent Gaussian special case and prove that the optimization problem has a closed-form minimizer. This means few-shot models can directly obtain an approximated minimizer without a complex optimization process. Furthermore, we also provide the accuracy bound $\widetilde{O}(1/n_{ta}+1/\sqrt{n_s})$ for the empirical solution, which still has better dependence on $n_{ta}$ compared to $n_s$. The results of the real-world experiments also show that the models obtained by only fine-tuning the encoder and decoder specific to the target distribution can produce novel images with the target feature, which supports our theoretical results.