  We consider the classical problem of estimating the covariance matrix of a
subgaussian distribution from i.i.d. samples in the novel context of coarse
quantization, i.e., instead of having full knowledge of the samples, they are
quantized to one or two bits per entry. This problem occurs naturally in signal
processing applications. We introduce new estimators in two different
quantization scenarios and derive non-asymptotic estimation error bounds in
terms of the operator norm. In the first scenario we consider a simple,
scale-invariant one-bit quantizer and derive an estimation result for the
correlation matrix of a centered Gaussian distribution. In the second scenario,
we add random dithering to the quantizer. In this case we can accurately
estimate the full covariance matrix of a general subgaussian distribution by
collecting two bits per entry of each sample. In both scenarios, our bounds
apply to masked covariance estimation. We demonstrate the near-optimality of
our error bounds by deriving corresponding (minimax) lower bounds and using
numerical simulations.
