DDPM-based generation of tilted samples

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 parameterized by $\theta \in {\mathbb{R}}^d$. We define a plug-in estimator and show that it is minimax-optimal. We develop an algorithm that, given original samples, generates exponentially tilted samples using a kerneled score estimator. The sample complexity of this algorithm is provably polynomial for bounded random variables, and avoids the curse of dimensionality. The proof uses tight Wasserstein bounds between the plug-in estimator distribution and the true distribution as a function of $\theta$ and $n$, and a novel score estimation bound. Furthermore, we also demonstrate sample complexity bounds for neural network-learned scores. Our theoretical results are supported by extensive simulations. Applications of our work include finance, weather and climate modeling, and many other domains, where the aim may be to generate samples from a tilted distribution that satisfies practically motivated moment constraints.