A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process. This quantity is especially relevant for procedures such as classifier guidance, which perturb the entire reverse trajectory rather than only the terminal sample. Prior analyses show that standard isotropic reverse covariances suffer an unavoidable $Ω(1/T)$ path-KL error as the number of denoising steps $T$ grows. We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to $O(1/T^2)$. To make full covariance matching practical, we introduce the Lanczos Gaussian sampler (LGS), a training-free, matrix-free method for sampling from the optimal reverse covariance using only covariance-vector products, which are available through Jacobian-vector products of the posterior mean. LGS avoids dense covariance storage and auxiliary covariance models. We prove that LGS approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product. Empirically, using only just three such steps improves sample quality over strong diagonal-covariance baselines, including OCM-DDPM, across standard image benchmarks. This identifies full covariance matching as both theoretically valuable and practically accessible for fast DDPM sampling.

翻译：在高斯型去噪扩散概率模型（DDPM）中，一个核心误差度量是精确反向链与学习得到的高斯反向过程之间的路径空间KL散度。该量对于分类器引导等过程尤为重要——这些过程扰动的是整个反向轨迹，而非仅终端样本。先前分析表明，随着去噪步数T增加，标准各向同性反向协方差会导致不可避免的$\Omega(1/T)$路径KL误差。本研究证明，匹配完整后验协方差可突破这一壁垒，实现阶次改进，将路径KL降至$O(1/T^2)$。为使完整协方差匹配具有实用性，我们提出Lanczos高斯采样器（LGS）——一种无需训练、免矩阵构建的方法，仅通过协方差-向量乘积（可通过后验均值的雅可比-向量乘积获得）从最优反向协方差中采样。LGS避免了稠密协方差存储和辅助协方差模型。我们证明，LGS近似误差随Lanczos步数呈指数衰减，而每步仅需一次雅可比-向量乘积。实验表明，在标准图像基准测试中，仅使用三步Lanczos即可在样本质量上超越包括OCM-DDPM在内的强对角线协方差基线方法。这证实了完整协方差匹配在理论上具有重要价值，且在实践中可实现快速DDPM采样。