High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization

In this paper, we propose a uniformly dithered one-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 three estimation problems: sparse covariance matrix estimation, sparse linear regression, and matrix completion. We study both sub-Gaussian and heavy-tailed regimes, with the underlying distribution of heavy-tailed data assumed to possess bounded second or fourth moment. For each model we propose new estimators based on one-bit quantized data. In sub-Gaussian regime, our estimators achieve optimal minimax rates up to logarithmic factors, which indicates that our quantization scheme nearly introduces no additional cost. In heavy-tailed regime, while the rates of our estimators become essentially slower, these results are either the first ones in such one-bit quantized and heavy-tailed setting, or exhibit some advantages over existing comparable results. More specifically, our results for one-bit compressed sensing feature generality of sensing vector (sub-Gaussian or even heavy-tailed) and tractable convex programming. A novel setting where both measurement and covariate are quantized is also first proposed and studied. For one-bit matrix completion, our method is essentially different from the standard likelihood approach and can handle pre-quantization random noise with unknown distribution. Experimental results on synthetic data are presented to support our theoretical analysis.