Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modeling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with the canonical choices currently made for diffusion models. For other distributions, however, we can improve upon these canonical choices, which we show both theoretically and empirically, by applying the algorithms to data distributions on manifolds and inspired by Bayesian inverse problems or simulation-based inference.

翻译：扩散模型对诸多应用领域产生了深远影响，包括那些数据本质上是无限维的领域，例如图像或时间序列。标准方法是先对数据进行离散化，然后将扩散模型应用于离散化后的数据。尽管这类方法在实践中具有吸引力，但所得算法的性能通常会随离散化参数的细化而恶化。本文中，我们直接构建了无限维空间中的扩散生成模型，并将其应用于函数的生成建模。我们证明了这些公式在无限维设定下是适定的，并给出了从样本到目标测度的与维度无关的距离界限。基于我们的理论，我们还为无限维扩散模型的设计制定了指导原则。对于图像分布，这些指导原则与当前扩散模型采用的经典选择一致。然而，对于其他分布，通过将算法应用于流形上的数据分布以及受贝叶斯逆问题或基于模拟的推断启发的分布，我们在理论和经验上均表明可以改进这些经典选择。