Denoising diffusions are a powerful method to generate approximate samples from high-dimensional data distributions. Recent results provide polynomial bounds on their convergence rate, assuming $L^2$-accurate scores. Until now, the tightest bounds were either superlinear in the data dimension or required strong smoothness assumptions. We provide the first convergence bounds which are linear in the data dimension (up to logarithmic factors) assuming only finite second moments of the data distribution. We show that diffusion models require at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps to approximate an arbitrary distribution on $\mathbb{R}^d$ corrupted with Gaussian noise of variance $\delta$ to within $\varepsilon^2$ in KL divergence. Our proof extends the Girsanov-based methods of previous works. We introduce a refined treatment of the error from discretizing the reverse SDE inspired by stochastic localization.

翻译：去噪扩散是一种强大的方法，可以从高维数据分布中生成近似样本。最近的结果提供了其收敛速度的多项式界，前提是$L^2$精确的得分函数。到目前为止，最紧的界要么在数据维度上呈超线性，要么需要强光滑性假设。我们首次给出了在数据分布仅具有有限二阶矩的假设下，收敛界与数据维度呈线性（至多对数因子）的结果。我们证明，扩散模型最多需要$\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$步，即可在KL散度上逼近$\mathbb{R}^d$上被方差为$\delta$的高斯噪声破坏的任意分布，误差控制在$\varepsilon^2$以内。我们的证明扩展了先前工作的基于Girsanov的方法。我们引入了一种受随机局部化启发的对逆SDE离散化误差的精细化处理。