We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

翻译：我们提出了函数扩散过程（FDPs），将基于分数的扩散模型推广到无限维函数空间。FDPs需要新的数学框架来描述前向和反向动力学，以及若干扩展以推导实用的训练目标。这些包括吉萨诺夫定理的无限维版本，以便能够计算证据下界（ELBO），以及抽样定理的无限维版本，以保证可数点集上的函数评估等价于无限维函数。我们使用FDPs构建了一类新的函数空间生成模型，这些模型无需专门的网络架构，且适用于任何类型的连续数据。在真实数据上的结果表明，FDPs采用简单的多层感知机（MLP）架构即可生成高质量图像，其参数数量比现有扩散模型少几个数量级。