  Denoising is intuitively related to projection. Indeed, under the manifold
hypothesis, adding random noise is approximately equivalent to orthogonal
perturbation. Hence, learning to denoise is approximately learning to project.
In this paper, we use this observation to interpret denoising diffusion models
as approximate gradient descent applied to the Euclidean distance function. We
then provide straight-forward convergence analysis of the DDIM sampler under
simple assumptions on the projection error of the denoiser. Finally, we propose
a new gradient-estimation sampler, generalizing DDIM using insights from our
theoretical results. In as few as 5-10 function evaluations, our sampler
achieves state-of-the-art FID scores on pretrained CIFAR-10 and CelebA models
and can generate high quality samples on latent diffusion models.
