Generating DDPM-based Samples from Tilted Distributions
This work addresses the need for generating samples from tilted distributions in domains like finance and climate modeling, but it appears incremental as it builds on existing diffusion methods with theoretical extensions.
The paper tackles the problem of generating diffusion-based samples from tilted distributions using a plug-in estimator, proving it is minimax-optimal and providing Wasserstein bounds and TV-accuracy results under assumptions, with support from simulations.
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.