从倾斜分布中生成扩散模型样本，理论保证高效可靠
Generating DDPM-based Samples from Tilted Distributions
- 用插件估计器实现对倾斜分布的最优逼近
- 在样本量n和倾斜参数θ下，给出输出与真实分布的Wasserstein距离界
- 适用于金融、气候等需满足矩约束的采样场景
给定来自d维概率分布的n个独立样本，目标是生成由原分布倾斜得到的新分布的扩散模型样本，倾斜程度由参数θ∈ℝᵈ控制。我们定义了一个插件估计器，并证明其为极小极大最优。推导了该估计器分布与真实分布之间的Wasserstein距离上界，随n和θ变化，揭示了输出与目标分布接近的条件。在若干假设下，还证明了对这些倾斜样本运行扩散模型具有总变差（TV）精度。理论结果通过大量模拟验证。本方法可用于金融、天气与气候建模等领域，实现满足实际矩约束的样本生成。
原文摘要 · Abstract (English)
Given $n$ independent samples from a $d$-dimensional probability distribution, our aim is to generate diffusion-based samples from a distribution obtained by tilting the original, where the degree of tilt is parametrized by $θ\in \mathbb{R}^d$. We define a plug-in estimator and show that it is minimax-optimal. We develop Wasserstein bounds between the distribution of the plug-in estimator and the true distribution as a function of $n$ and $θ$, illustrating regimes where the output and the desired true distribution are close. Further, under some assumptions, we prove the TV-accuracy of running Diffusion on these tilted samples. Our theoretical results are supported by extensive simulations. Applications of our work include finance, weather and climate modelling, and many other domains, where the aim may be to generate samples from a tilted distribution that satisfies practically motivated moment constraints.
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