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.

翻译：低秩多元回归是一种重要的统计学习模型，它将高度相关的任务整合为带有系数矩阵低秩先验的多响应回归问题。本文研究了量化低秩回归，即响应变量和/或协变量被离散化为有限精度的实际场景。我们聚焦于底层系数矩阵的估计。为构建能实现任意小误差的一致估计量，我们采用带随机抖动的均匀量化策略——在量化前向数据添加适当的随机噪声。具体而言，对响应变量使用均匀抖动，对协变量使用三角抖动。基于量化数据，我们提出了约束Lasso和正则化Lasso估计量，并推导了非渐近误差界。借助抖动机制，估计量达到了极小极大最优速率，而量化仅轻微增加了误差率中的乘法因子。此外，我们将结果扩展至具有矩阵响应的低秩回归模型。通过合成数据与图像恢复的仿真实验，我们验证并展示了理论结果。