Estimation of a vector from quantized linear measurements is a common problem
for which simple linear techniques are suboptimal -- sometimes greatly so. This
paper develops generalized approximate message passing (GAMP) algorithms for
minimum mean-squared error estimation of a random vector from quantized linear
measurements, notably allowing the linear expansion to be overcomplete or
undercomplete and the scalar quantization to be regular or non-regular. GAMP is
a recently-developed class of algorithms that uses Gaussian approximations in
belief propagation and allows arbitrary separable input and output channels.
Scalar quantization of measurements is incorporated into the output channel
formalism, leading to the first tractable and effective method for
high-dimensional estimation problems involving non-regular scalar quantization.
Non-regular quantization is empirically demonstrated to greatly improve
rate-distortion performance in some problems with oversampling or with
undersampling combined with a sparsity-inducing prior. Under the assumption of
a Gaussian measurement matrix with i.i.d. entries, the asymptotic error
performance of GAMP can be accurately predicted and tracked through the state
evolution formalism. We additionally use state evolution to design MSE-optimal
scalar quantizers for GAMP signal reconstruction and empirically demonstrate
the superior error performance of the resulting quantizers.Comment: 12 pages, 8 figure
