  We present SURE-Score: an approach for learning score-based generative models
using training samples corrupted by additive Gaussian noise. When a large
training set of clean samples is available, solving inverse problems via
score-based (diffusion) generative models trained on the underlying
fully-sampled data distribution has recently been shown to outperform
end-to-end supervised deep learning. In practice, such a large collection of
training data may be prohibitively expensive to acquire in the first place. In
this work, we present an approach for approximately learning a score-based
generative model of the clean distribution, from noisy training data. We
formulate and justify a novel loss function that leverages Stein's unbiased
risk estimate to jointly denoise the data and learn the score function via
denoising score matching, while using only the noisy samples. We demonstrate
the generality of SURE-Score by learning priors and applying posterior sampling
to ill-posed inverse problems in two practical applications from different
domains: compressive wireless multiple-input multiple-output channel estimation
and accelerated 2D multi-coil magnetic resonance imaging reconstruction, where
we demonstrate competitive reconstruction performance when learning at
signal-to-noise ratio values of 0 and 10 dB, respectively.
