  Low-rank multivariate regression (LRMR) is an important statistical learning
model that combines highly correlated tasks as a multiresponse regression
problem with low-rank priori on the coefficient matrix. In this paper, we study
quantized LRMR, a practical setting where the responses and/or the covariates
are discretized to finite precision. We focus on the estimation of the
underlying coefficient matrix. To make consistent estimator that could achieve
arbitrarily small error possible, we employ uniform quantization with random
dithering, i.e., we add appropriate random noise to the data before
quantization. Specifically, uniform dither and triangular dither are used for
responses and covariates, respectively. Based on the quantized data, we propose
the constrained Lasso and regularized Lasso estimators, and derive the
non-asymptotic error bounds. With the aid of dithering, the estimators achieve
minimax optimal rate, while quantization only slightly worsens the
multiplicative factor in the error rate. Moreover, we extend our results to a
low-rank regression model with matrix responses. We corroborate and demonstrate
our theoretical results via simulations on synthetic data or image restoration.
