Message-Passing De-Quantization with Applications to Compressed Sensing
U.S. Kamilov, V.K. Goyal, S. Rangan
IEEE Transactions on Signal Processing, vol. 60, no. 12, pp. 6270–6281, December 2012.
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 message-passing de-quantization (MPDQ) 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. The algorithm is based on generalized approximate message passing (GAMP), a recently-developed Gaussian approximation of loopy belief propagation for estimation with linear transforms and nonlinear componentwise-separable output channels. For MPDQ, 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. The algorithm is computationally simple and can incorporate arbitrary separable priors on the input vector including sparsity-inducing priors that arise in the context of compressed sensing. Moreover, under the assumption of a Gaussian measurement matrix with i.i.d. entries, the asymptotic error performance of MPDQ can be accurately predicted and tracked through a simple set of scalar state evolution equations. We additionally use state evolution to design MSE-optimal scalar quantizers for MPDQ signal reconstruction and empirically demonstrate the superior error performance of the resulting quantizers. In particular, our results show that non-regular quantization can greatly improve rate-distortion performance in some problems with oversampling or with undersampling combined with a sparsity-inducing prior.
@ARTICLE(http://bigwww.epfl.ch/publications/kamilov1205.html,
AUTHOR="Kamilov, U.S. and Goyal, V.K. and Rangan, S.",
TITLE="Message-Passing De-Quantization with Applications to Compressed
Sensing",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2012",
volume="60",
number="12",
pages="6270--6281",
month="December",
note="")