Linear signal recovery from $b$-bit-quantized linear measurements:   precise analysis of the trade-off between bit depth and number of   measurements
  We consider the problem of recovering a high-dimensional 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 non-linearity 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 one-bit observations due to Plan and Vershynin [\emph{IEEE~Trans. Inform. Theory, \textbf{59} (2013), 482--494}]. At the heart of the present paper is a precise characterization of the optimal trade-off 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, Lloyd-Max 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 non-linearity resulting from quantization into account. The results of that comparison point to a regime change depending on the noise level: in a low-noise setting, linear signal recovery falls short of more sophisticated competitors while being competitive in moderate- and high-noise settings. 
Martin Slawski and Ping Li
cs.IT math.IT stat.ME