  Score-based diffusion models (SBDM) have recently emerged as state-of-the-art
approaches for image generation. Existing SBDMs are typically formulated in a
finite-dimensional setting, where images are considered as tensors of finite
size. This paper develops SBDMs in the infinite-dimensional setting, that is,
we model the training data as functions supported on a rectangular domain.
Besides the quest for generating images at ever higher resolution, our primary
motivation is to create a well-posed infinite-dimensional learning problem so
that we can discretize it consistently on multiple resolution levels. We
thereby intend to obtain diffusion models that generalize across different
resolution levels and improve the efficiency of the training process. We
demonstrate how to overcome two shortcomings of current SBDM approaches in the
infinite-dimensional setting. First, we modify the forward process to ensure
that the latent distribution is well-defined in the infinite-dimensional
setting using the notion of trace class operators. We derive the reverse
processes for finite approximations. Second, we illustrate that approximating
the score function with an operator network is beneficial for multilevel
training. After deriving the convergence of the discretization and the
approximation of multilevel training, we implement an infinite-dimensional SBDM
approach and show the first promising results on MNIST and Fashion-MNIST,
underlining our developed theory.
