Diffusion generative models have recently been applied to domains where the available data can be seen as a discretization of an underlying function, such as audio signals or time series. However, these models operate directly on the discretized data, and there are no semantics in the modeling process that relate the observed data to the underlying functional forms. We generalize diffusion models to operate directly in function space by developing the foundational theory for such models in terms of Gaussian measures on Hilbert spaces. A significant benefit of our function space point of view is that it allows us to explicitly specify the space of functions we are working in, leading us to develop methods for diffusion generative modeling in Sobolev spaces. Our approach allows us to perform both unconditional and conditional generation of function-valued data. We demonstrate our methods on several synthetic and real-world benchmarks.

翻译：扩散生成模型近期已被应用于可将可用数据视为潜在函数离散化表示的领域，例如音频信号或时间序列。然而，这些模型直接对离散化数据进行操作，其建模过程中缺乏将观测数据与潜在函数形式相关联的语义机制。我们通过发展基于希尔伯特空间上高斯测度的基本理论，将扩散模型推广为直接在函数空间中进行运算。函数空间视角的一个显著优势在于能够显式指定所处理的函数空间，进而发展出索伯列夫空间中的扩散生成建模方法。我们的方法能够对函数值数据执行无条件生成与条件生成，并在多个合成基准与真实世界基准上验证了其有效性。