Quantized Low-Rank Multivariate Regression with Random Dithering

TL;DR

This paper introduces a method for low-rank multivariate regression with quantized data, employing random dithering to achieve minimax optimal error bounds, and extends the approach to matrix responses with demonstrated effectiveness.

Contribution

It proposes a novel quantized LRMR approach using random dithering and constrained Lasso estimators, achieving near-optimal error rates in discretized data settings.

Findings

Estimators achieve minimax optimal error rates with dithering.

Quantization slightly increases the error multiplicative factor.

Method extends to low-rank matrix response models.

Abstract

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.