# Quantized Low-Rank Multivariate Regression with Random Dithering

arXiv: https://arxiv.org/abs/2302.11197 · HTML: https://mlsanity.com/paper/2302.11197/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 81 references that are in the map.
- First posted on arXiv 22 Feb 2023.

## Builds on (most cited first, 20 of 20)

- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [Federated Learning: Strategies for Improving Communication Efficiency](https://mlsanity.com/paper/1610.05492/) (2016)
- [DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients](https://mlsanity.com/paper/1606.06160/) (2016)
- [Quantized Neural Networks: Training Neural Networks with Low Precision Weights and Activations](https://mlsanity.com/paper/1609.07061/) (2016)
- [FedPAQ: A Communication-Efficient Federated Learning Method with Periodic Averaging and Quantization](https://mlsanity.com/paper/1909.13014/) (2019)
- [Decentralized Stochastic Optimization and Gossip Algorithms with Compressed Communication](https://mlsanity.com/paper/1902.00340/) (2019)
- [UVeQFed: Universal Vector Quantization for Federated Learning](https://mlsanity.com/paper/2006.03262/) (2020)
- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [A Max-Norm Constrained Minimization Approach to 1-Bit Matrix Completion](https://mlsanity.com/paper/1309.6013/) (2013)
- [A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery](https://mlsanity.com/paper/1603.08315/) (2016)
- [High Dimensional Statistical Estimation under Uniformly Dithered One-bit Quantization](https://mlsanity.com/paper/2202.13157/) (2022)
- [The Generalized Lasso for Sub-gaussian Measurements with Dithered Quantization](https://mlsanity.com/paper/1807.06976/) (2018)
- [Adaptive Multinomial Matrix Completion](https://mlsanity.com/paper/1408.6218/) (2014)
- [On Low-rank Trace Regression under General Sampling Distribution](https://mlsanity.com/paper/1904.08576/) (2019)
- [Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery](https://mlsanity.com/paper/2212.14562/) (2022)
- [Robust Reduced Rank Regression](https://mlsanity.com/paper/1509.03938/) (2015)
- [Quantizing data for distributed learning](https://mlsanity.com/paper/2012.07913/) (2020)
- [Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss](https://mlsanity.com/paper/2202.02063/) (2022)
- [Covariance Estimation for Matrix-valued Data](https://mlsanity.com/paper/2004.05281/) (2020)
- [Non-Convex Structured Phase Retrieval](https://mlsanity.com/paper/2006.13298/) (2020)

## Built on by (most cited first, 5 of 5)

- [A Parameter-Free Two-Bit Covariance Estimator with Improved Operator Norm Error Rate](https://mlsanity.com/paper/2308.16059/) (2023)
- [Bit Efficient Toeplitz Covariance Estimation](https://mlsanity.com/paper/2412.12678/) (2024)
- [Compressive Toeplitz Covariance Estimation From Few-Bit Quantized Measurements With Applications to DOA Estimation](https://mlsanity.com/paper/2512.22527/) (2025)
- [Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation](https://mlsanity.com/paper/2602.12982/) (2026)
- [Understanding the Weight Averaging Mechanism in LLM Training for Post-Training Quantization](https://mlsanity.com/paper/2610.05329/) (2026)

## Abstract (arXiv)

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.