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.

翻译：给定来自$d$维概率分布的$n$个独立样本，我们的目标是生成由原始分布倾斜（倾斜程度由参数$\theta\in \mathbb{R}^d$控制）后得到的分布中的扩散模型样本。我们定义了一个插件估计量，并证明其具有极小极大最优性。我们建立了该插件估计量分布与真实分布之间的Wasserstein界（该界是$n$和$\theta$的函数），阐明了输出分布与期望真实分布接近的区间。此外，在某些假设下，我们证明了在这些倾斜样本上运行扩散过程的TV精度。我们的理论结果得到了大量模拟实验的支持。本研究的应用涵盖金融、天气与气候建模以及许多其他领域，其目标可能是从满足实际矩约束的倾斜分布中生成样本。