标题： 无维度收敛的扩散模型用于近似高斯混合物 显示英文标题

标题： Dimension-Free Convergence of Diffusion Models for Approximate Gaussian Mixtures

摘要： 扩散模型因其卓越的生成性能而脱颖而出，特别是在通过迭代去噪来生成高质量样本方面表现尤为突出。虽然当前理论表明，为了准确生成样本所需的去噪步数应该随数据维度线性增长，但这并不能反映广泛使用的算法（如去噪扩散概率模型 DDPM）的实际效率。 本文研究了扩散模型在从可以通过高斯混合模型（GMM）良好近似的复杂高维分布中采样的有效性。 对于这些分布，我们的主要结果表明，DDPM 最多需要$\widetilde{O}(1/\varepsilon)$次迭代即可达到总变差（TV）距离下的$\varepsilon$-精确分布，且这一结果独立于环境维度$d$和分量数量$K$，至多以对数因子依赖。 此外，这一结果对评分估计误差具有鲁棒性。 这些发现强调了扩散模型在高维设置下的显著有效性，考虑到 GMM 的通用逼近能力，同时为它们在实际中的成功提供了理论见解。 显示英文摘要

摘要： Diffusion models are distinguished by their exceptional generative performance, particularly in producing high-quality samples through iterative denoising. While current theory suggests that the number of denoising steps required for accurate sample generation should scale linearly with data dimension, this does not reflect the practical efficiency of widely used algorithms like Denoising Diffusion Probabilistic Models (DDPMs). This paper investigates the effectiveness of diffusion models in sampling from complex high-dimensional distributions that can be well-approximated by Gaussian Mixture Models (GMMs). For these distributions, our main result shows that DDPM takes at most $\widetilde{O}(1/\varepsilon)$ iterations to attain an $\varepsilon$-accurate distribution in total variation (TV) distance, independent of both the ambient dimension $d$ and the number of components $K$, up to logarithmic factors. Furthermore, this result remains robust to score estimation errors. These findings highlight the remarkable effectiveness of diffusion models in high-dimensional settings given the universal approximation capability of GMMs, and provide theoretical insights into their practical success.