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 a finite size. This papers 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 hope 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. Second, we illustrate that approximating the score function with an operator network, in our case Fourier neural operators (FNOs), is beneficial for multilevel training. After deriving the forward and reverse process in the infinite-dimensional setting, we show their well-posedness, derive adequate discretizations, and investigate the role of the latent distributions. We provide first promising numerical results on two datasets, MNIST and material structures. In particular, we show that multilevel training is feasible within this framework.
翻译:基于分数的传播模型(SBDM)最近成为了图像生成的最先进方法。现有的SBDMM(SBDM)通常是在有限的维度环境中制作的,其图像被视为有限尺寸的分母。这些论文在无限维度环境中开发SBDM(SBDM),即我们用在矩形域上支持的功能来模拟培训数据。除了寻求以更高分辨率生成图像外,我们的主要动机是制造一个有广泛位置的无限范围学习问题,以便我们得以在多个分辨率水平上始终分解。我们希望获得一个分布在不同分辨率水平上的推广模型,提高培训进程的效率。我们展示了如何克服当前SBDMDM方法在无限维度环境中的两个缺点。首先,我们用追踪类操作者的概念来修改前方和后方程序以确保在无限的维度环境中对潜在分布进行明确界定。第二,我们证明与操作者网络匹配的分数功能,在我们的例子中,四线操作者(FNOs),有利于多层次的培训。我们展示了在无限维的设置前方和反向方向结构中,我们展示了它们有希望的数值结构。我们要的多层次结构。我们展示了这种数字结构。我们很好地展示了它们。我们进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进进</s>