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We consider the problem of recovering a high-dimensional structured signalfrom 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 theobservations are treated as if they arose from a linear measurement model.Specifically, the focus is on a generalization of a method for one-bitobservations due to Plan and Vershynin [\emph{IEEE~Trans. Inform. Theory,\textbf{59} (2013), 482--494}]. At the heart of the present paper is a precisecharacterization of the optimal trade-off between the number of measurements$m$ and the bit depth per measurement $b$ given a total budget of $B = m \cdotb$ bits when the goal is to minimize the $\ell_2$-error in estimating thesignal. It turns out that the choice $b = 1$ is optimal for estimating the unitvector (direction) corresponding to the signal for any level of additiveGaussian noise before quantization as well as for a specific model ofadversarial noise, while the choice $b = 2$ is optimal for estimating thedirection and the norm (scale) of the signal. Moreover, Lloyd-Max quantizationis shown to be an optimal quantization scheme w.r.t. $\ell_2$-estimation error.Our analysis is corroborated by numerical experiments showing nearly perfectagreement with our theoretical predictions. The paper is complemented by anempirical comparison to alternative methods of signal recovery taking thenon-linearity resulting from quantization into account. The results of thatcomparison point to a regime change depending on the noise level: in alow-noise setting, linear signal recovery falls short of more sophisticatedcompetitors while being competitive in moderate- and high-noise settings.

Linear signal recovery from $b$-bit-quantized linear measurements: precise analysis of the trade-off between bit depth and number of measurements