  In the problem of structured signal recovery from high-dimensional linear
observations, it is commonly assumed that full-precision measurements are
available. Under this assumption, the recovery performance of the popular
Generalized Lasso (G-Lasso) is by now well-established. In this paper, we
extend these types of results to the practically relevant settings with
quantized measurements. We study two extremes of the quantization schemes,
namely, uniform and one-bit quantization; the former imposes no limit on the
number of quantization bits, while the second only allows for one bit. In the
presence of a uniform dithering signal and when measurement vectors are
sub-gaussian, we show that the same algorithm (i.e., the G-Lasso) has favorable
recovery guarantees for both uniform and one-bit quantization schemes. Our
theoretical results, shed light on the appropriate choice of the range of
values of the dithering signal and accurately capture the error dependence on
the problem parameters. For example, our error analysis shows that the G-Lasso
with one-bit uniformly dithered measurements leads to only a logarithmic rate
loss compared to the full-precision measurements.
