扩散拟蒙特卡洛
Diffusion Quasi-Monte Carlo
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中文总结 AI 辅助
该研究提出结合扩散输运映射与随机化拟蒙特卡洛的高维积分方法，给出理论收敛保证，实验验证其在高维任务中可在成本不变的同时降低精度指标的随机化标准差。
中文摘要 AI 辅助
我们研究了基于扩散输运映射和随机化拟蒙特卡洛（RQMC，randomized quasi-Monte Carlo）的复杂目标测度高维数值积分问题。基于分数的扩散模型会诱导出一条确定性概率流常微分方程（ODE），可将简单先验分布输运至目标分布，这为将单位立方体上的低偏差点转化为有信息的样本提供了一种有理论依据的方法。我们通过将高斯基变换（逐分量逆高斯累积分布函数CDF）与欧拉离散化的概率流ODE相组合，构建了一个从立方体到目标的映射。为在输运近似下保持无偏性，我们将积分表述为立方体上的重要性采样（IS，importance sampling）。我们的主要结果给出了可验证的条件，在这些条件下，所得的IS被积函数满足边界增长条件，意味着置乱网格的均方根误差（RMSE）为$O(N^{-1+ε})$。随后我们证明，在学习到的向量场满足温和的有界导数假设下，扩散概率流输运满足这些条件，并显式控制了逆高斯CDF引入的边界奇异性。实验涵盖从二维混合分布到784维图像以及40960维条件涡度同化任务；在最后一项任务中，分块置乱Sobol采样在基本不改变在线去噪成本的情况下，降低了非线性精度指标的随机化标准差。这些结果共同为结合扩散生成建模与高精度RQMC积分提供了理论和实证基础。
英文摘要
We study high-dimensional numerical integration with respect to complex target measures using diffusion-based transport maps and randomized quasi-Monte Carlo (RQMC). Score-based diffusion models induce a deterministic probability flow ODE that transports a simple prior to the target, suggesting a principled way to transform low-discrepancy points on the unit cube into informative samples. We construct a cube-to-target map by composing a Gaussian base transformation (the component-wise inverse Gaussian CDF) with an Euler-discretized probability flow ODE. To retain unbiasedness under transport approximation, we formulate integration as importance sampling (IS) on the cube. Our main result provides verifiable conditions under which the resulting IS integrand satisfies the boundary growth condition, implying an $O(N^{-1+ε})$ RMSE for scrambled nets. We then establish these conditions for diffusion probability-flow transport under mild bounded-derivative assumptions on the learned vector field, explicitly controlling the boundary singularities introduced by the inverse Gaussian CDF. Experiments range from a 2D mixture to 784D images and a 40,960D conditional vorticity-assimilation task; in the latter, blocked scrambled Sobol' sampling reduces the randomization standard deviation of nonlinear accuracy metrics at essentially unchanged online denoising cost. Together, these results give a theoretical and empirical foundation for combining diffusion generative modeling with high-precision RQMC integration.
