用随机微分方程引导矩约束，实现高效高维最大熵采样。
MGD: Moment Guided Diffusion for Maximum Entropy Generation
- 基于随机插值框架，通过解SDE在有限时间内匹配指定矩
- 大噪声极限下收敛到最大熵分布，可直接计算动态熵估计
- 适用于金融、湍流、宇宙场等多尺度高维过程的负熵估计
从有限信息生成样本是多个科学领域的基础问题。经典最大熵方法虽能提供基于矩约束的不确定性量化，但依赖MCMC或Langevin动力学采样，在高维下通常出现指数级缓慢。而基于扩散和流匹配的生成模型虽能高效将噪声转化为数据，但理论保障不足，且在数据稀缺时易过拟合。本文提出矩引导扩散（MGD），融合两类方法优势。基于随机插值框架，MGD通过求解一个随机微分方程，在有限时间内引导过程矩趋近于预设值，从而避免基于平衡态方法的慢混合问题。我们在大噪声极限下严格证明了MGD收敛至最大熵分布，并推导出可直接从动态中计算的熵估计量。应用于金融时间序列、湍流流场及宇宙场，结合小波散射矩，实现了对高维多尺度过程的负熵估计。
原文摘要 · Abstract (English)
Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.
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