Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction

Diffusion reconstruction plays a critical role in various applications such as image editing, restoration, and style transfer. In theory, the reconstruction should be simple - it just inverts and regenerates images by numerically solving the Probability Flow-Ordinary Differential Equation (PF-ODE). Yet in practice, noticeable reconstruction errors have been observed, which cannot be well explained by numerical errors. In this work, we identify a deeper intrinsic property in the PF-ODE generation process, the instability, that can further amplify the reconstruction errors. The root of this instability lies in the sparsity inherent in the generation distribution, which means that the probability is concentrated on scattered and small regions while the vast majority remains almost empty. To demonstrate the existence of instability and its amplification on reconstruction error, we conduct experiments on both toy numerical examples and popular open-sourced diffusion models. Furthermore, based on the characteristics of image data, we theoretically prove that the instability's probability converges to one as the data dimensionality increases. Our findings highlight the inherent challenges in diffusion-based reconstruction and can offer insights for future improvements.