Large denoising diffusion models, such as Stable Diffusion, have been trained on billions of image-caption pairs to perform text-conditioned image generation. As a byproduct of this training, these models have acquired general knowledge about image statistics, which can be useful for other inference tasks. However, when confronted with sampling an image under new constraints, e.g. generating the missing parts of an image, using large pre-trained text-to-image diffusion models is inefficient and often unreliable. Previous approaches either utilized backpropagation through the denoiser network, making them significantly slower and more memory-demanding than simple text-to-image generation, or only enforced the constraint locally, failing to capture critical long-range correlations in the sampled image. In this work, we propose an algorithm that enables fast, high-quality generation under arbitrary constraints. We show that in denoising diffusion models, we can employ an approximation to Newton's optimization method that allows us to speed up inference and avoid the expensive backpropagation operations. Our approach produces results that rival or surpass the state-of-the-art training-free inference methods while requiring a fraction of the time. We demonstrate the effectiveness of our algorithm under both linear (inpainting, super-resolution) and non-linear (style-guided generation) constraints. An implementation is provided at https://github.com/cvlab-stonybrook/fast-constrained-sampling.

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