A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is obtained by solving differential equations (DE) defined by the learnt model, a process which has shown to be prohibitively slow. Numerous efforts on speeding-up this process have consisted on crafting powerful ODE solvers. Despite being quick, such solvers do not usually reach the optimal quality achieved by available slow SDE solvers. Our goal is to propose SDE solvers that reach optimal quality without requiring several hundreds or thousands of NFEs to achieve that goal. We propose Stochastic Explicit Exponential Derivative-free Solvers (SEEDS), improving and generalizing Exponential Integrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exact solutions of diffusion SDEs, we craft SEEDS to analytically compute the linear part of such solutions. Inspired by the Exponential Time-Differencing method, SEEDS use a novel treatment of the stochastic components of solutions, enabling the analytical computation of their variance, and contains high-order terms allowing to reach optimal quality sampling $\sim3$-$5\times$ faster than previous SDE methods. We validate our approach on several image generation benchmarks, showing that SEEDS outperform or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.

翻译：一类名为扩散概率模型（DPMs）的强大生成模型已崭露头角。前向扩散过程逐步向数据添加噪声，而模型则学习逐步去噪。从预训练DPMs中采样需通过求解由学习模型定义的微分方程（DE）实现，该过程已被证明速度极慢。为加速此过程，众多研究致力于构建高效的ODE求解器。尽管这些求解器速度快，但通常无法达到现有慢速SDE求解器的最优质量。我们的目标是提出能在不要求数百或数千次NFE的前提下达到最优质量的SDE求解器。我们提出随机显式指数无导数求解器（SEEDS），在多个框架上改进并推广了指数积分器方法至随机情形。通过深入分析扩散SDE精确解的表达形式，我们设计了SEEDS以解析计算此类解的线性部分。受指数时间差分方法启发，SEEDS采用对解中随机分量的新型处理方法，实现对其方差的解析计算，并包含高阶项，从而能够比现有SDE方法快约3-5倍达到最优质量采样。我们在多个图像生成基准上验证了该方法，表明SEEDS优于或与现有SDE求解器竞争。与后者不同，SEEDS无需导数和训练，并且我们为其提供了强收敛保证的完整证明。