  We consider the general problem of recovering a high-dimensional signal from
noisy quantized measurements. Quantization, especially coarse quantization such
as 1-bit sign measurements, leads to severe information loss and thus a good
prior knowledge of the unknown signal is helpful for accurate recovery.
Motivated by the power of score-based generative models (SGM, also known as
diffusion models) in capturing the rich structure of natural signals beyond
simple sparsity, we propose an unsupervised data-driven approach called
quantized compressed sensing with SGM (QCS-SGM), where the prior distribution
is modeled by a pre-trained SGM. To perform posterior sampling, an annealed
pseudo-likelihood score called noise perturbed pseudo-likelihood score is
introduced and combined with the prior score of SGM. The proposed QCS-SGM
applies to an arbitrary number of quantization bits. Experiments on a variety
of baseline datasets demonstrate that the proposed QCS-SGM significantly
outperforms existing state-of-the-art algorithms by a large margin for both
in-distribution and out-of-distribution samples. Moreover, as a posterior
sampling method, QCS-SGM can be easily used to obtain confidence intervals or
uncertainty estimates of the reconstructed results. The code is available at
https://github.com/mengxiangming/QCS-SGM.
