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：我们提出泛函扩散过程（FDPs），它将基于得分的扩散模型推广至无限维函数空间。FDPs需要新的数学框架来描述正向与反向动力学，并需要若干扩展来推导实用的训练目标。这些扩展包括：用于计算ELBO的无限维版本Girsanov定理，以及用于保证在可数点集上的泛函评估等价于无限维函数的采样定理。我们利用FDPs在函数空间中构建新型生成模型，该模型无需特定网络架构，可处理任意类型的连续数据。在真实数据上的实验结果表明，FDPs通过简单的MLP架构即可实现高质量图像生成，其参数量比现有扩散模型少几个数量级。