  Ambient diffusion is a recently proposed framework for training diffusion
models using corrupted data. Both Ambient Diffusion and alternative SURE-based
approaches for learning diffusion models from corrupted data resort to
approximations which deteriorate performance. We present the first framework
for training diffusion models that provably sample from the uncorrupted
distribution given only noisy training data, solving an open problem in this
space. Our key technical contribution is a method that uses a double
application of Tweedie's formula and a consistency loss function that allows us
to extend sampling at noise levels below the observed data noise. We also
provide further evidence that diffusion models memorize from their training
sets by identifying extremely corrupted images that are almost perfectly
reconstructed, raising copyright and privacy concerns. Our method for training
using corrupted samples can be used to mitigate this problem. We demonstrate
this by fine-tuning Stable Diffusion XL to generate samples from a distribution
using only noisy samples. Our framework reduces the amount of memorization of
the fine-tuning dataset, while maintaining competitive performance.
