Guidance with Spherical Gaussian Constraint for Conditional Diffusion

Recent advances in diffusion models attempt to handle conditional generative tasks by utilizing a differentiable loss function for guidance without the need for additional training. While these methods achieved certain success, they often compromise on sample quality and require small guidance step sizes, leading to longer sampling processes. This paper reveals that the fundamental issue lies in the manifold deviation during the sampling process when loss guidance is employed. We theoretically show the existence of manifold deviation by establishing a certain lower bound for the estimation error of the loss guidance. To mitigate this problem, we propose Diffusion with Spherical Gaussian constraint (DSG), drawing inspiration from the concentration phenomenon in high-dimensional Gaussian distributions. DSG effectively constrains the guidance step within the intermediate data manifold through optimization and enables the use of larger guidance steps. Furthermore, we present a closed-form solution for DSG denoising with the Spherical Gaussian constraint. Notably, DSG can seamlessly integrate as a plugin module within existing training-free conditional diffusion methods. Implementing DSG merely involves a few lines of additional code with almost no extra computational overhead, yet it leads to significant performance improvements. Comprehensive experimental results in various conditional generation tasks validate the superiority and adaptability of DSG in terms of both sample quality and time efficiency.