Linear signal recovery from $b$bitquantized linear measurements: precise analysis of the tradeoff between bit depth and number of measurements
Abstract
We consider the problem of recovering a highdimensional structured signal from independent Gaussian linear measurements each of which is quantized to $b$ bits. Our interest is in linear approaches to signal recovery, where "linear" means that nonlinearity resulting from quantization is ignored and the observations are treated as if they arose from a linear measurement model. Specifically, the focus is on a generalization of a method for onebit observations due to Plan and Vershynin [\emph{IEEE~Trans. Inform. Theory, \textbf{59} (2013), 482494}]. At the heart of the present paper is a precise characterization of the optimal tradeoff between the number of measurements $m$ and the bit depth per measurement $b$ given a total budget of $B = m \cdot b$ bits when the goal is to minimize the $\ell_2$error in estimating the signal. It turns out that the choice $b = 1$ is optimal for estimating the unit vector (direction) corresponding to the signal for any level of additive Gaussian noise before quantization as well as for a specific model of adversarial noise, while the choice $b = 2$ is optimal for estimating the direction and the norm (scale) of the signal. Moreover, LloydMax quantization is shown to be an optimal quantization scheme w.r.t. $\ell_2$estimation error. Our analysis is corroborated by numerical experiments showing nearly perfect agreement with our theoretical predictions. The paper is complemented by an empirical comparison to alternative methods of signal recovery taking the nonlinearity resulting from quantization into account. The results of that comparison point to a regime change depending on the noise level: in a lownoise setting, linear signal recovery falls short of more sophisticated competitors while being competitive in moderate and highnoise settings.
 Publication:

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.02649
 Bibcode:
 2016arXiv160702649S
 Keywords:

 Computer Science  Information Theory;
 Statistics  Methodology