  In this paper, we propose a uniformly dithered 1-bit quantization scheme for
high-dimensional statistical estimation. The scheme contains truncation,
dithering, and quantization as typical steps. As canonical examples, the
quantization scheme is applied to the estimation problems of sparse covariance
matrix estimation, sparse linear regression (i.e., compressed sensing), and
matrix completion. We study both sub-Gaussian and heavy-tailed regimes, where
the underlying distribution of heavy-tailed data is assumed to have bounded
moments of some order. We propose new estimators based on 1-bit quantized data.
In sub-Gaussian regime, our estimators achieve near minimax rates, indicating
that our quantization scheme costs very little. In heavy-tailed regime, while
the rates of our estimators become essentially slower, these results are either
the first ones in an 1-bit quantized and heavy-tailed setting, or already
improve on existing comparable results from some respect. Under the
observations in our setting, the rates are almost tight in compressed sensing
and matrix completion. Our 1-bit compressed sensing results feature general
sensing vector that is sub-Gaussian or even heavy-tailed. We also first
investigate a novel setting where both the covariate and response are
quantized. In addition, our approach to 1-bit matrix completion does not rely
on likelihood and represent the first method robust to pre-quantization noise
with unknown distribution. Experimental results on synthetic data are presented
to support our theoretical analysis.
