  Designing efficient sparse recovery algorithms that could handle noisy
quantized measurements is important in a variety of applications -- from radar
to source localization, spectrum sensing and wireless networking. We take
advantage of the approximate message passing (AMP) framework to achieve this
goal given its high computational efficiency and state-of-the-art performance.
In AMP, the signal of interest is assumed to follow certain prior distribution
with unknown parameters. Previous works focused on finding the parameters that
maximize the measurement likelihood via expectation maximization -- an
increasingly difficult problem to solve in cases involving complicated
probability models. In this paper, we treat the parameters as unknown variables
and compute their posteriors via AMP. The parameters and signal of interest can
then be jointly recovered. Compared to previous methods, the proposed approach
leads to a simple and elegant parameter estimation scheme, allowing us to
directly work with 1-bit quantization noise model. We then further extend our
approach to general multi-bit quantization noise model. Experimental results
show that the proposed framework provides significant improvement over
state-of-the-art methods across a wide range of sparsity and noise levels.
