# Robust Reduced Rank Regression

arXiv: https://arxiv.org/abs/1509.03938 · HTML: https://mlsanity.com/paper/1509.03938/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2015.

## Built on by (most cited first, 8 of 8)

- [Gaining Outlier Resistance with Progressive Quantiles: Fast Algorithms and Theoretical Studies](https://mlsanity.com/paper/2112.08471/) (2021)
- [Analysis of Generalized Bregman Surrogate Algorithms for Nonsmooth Nonconvex Statistical Learning](https://mlsanity.com/paper/2112.09191/) (2021)
- [Quantized Low-Rank Multivariate Regression with Random Dithering](https://mlsanity.com/paper/2302.11197/) (2023)
- [Robust Regularized Low-Rank Matrix Models for Regression and Classification](https://mlsanity.com/paper/2205.07106/) (2022)
- [Slow Kill for Big Data Learning](https://mlsanity.com/paper/2305.01726/) (2023)
- [Simultaneous Heterogeneity and Reduced-rank Learning for Multivariate Response Regression](https://mlsanity.com/paper/2512.06514/) (2025)
- [Mining the Factor Zoo: Estimation of Latent Factor Models with Sufficient Proxies](https://mlsanity.com/paper/2212.12845/) (2022)
- [Higher Order Reduced Rank Regression](https://mlsanity.com/paper/2503.06528/) (2025)

## Abstract (arXiv)

In high-dimensional multivariate regression problems, enforcing low rank in the coefficient matrix offers effective dimension reduction, which greatly facilitates parameter estimation and model interpretation. However, commonly-used reduced-rank methods are sensitive to data corruption, as the low-rank dependence structure between response variables and predictors is easily distorted by outliers. We propose a robust reduced-rank regression approach for joint modeling and outlier detection. The problem is formulated as a regularized multivariate regression with a sparse mean-shift parametrization, which generalizes and unifies some popular robust multivariate methods. An efficient thresholding-based iterative procedure is developed for optimization. We show that the algorithm is guaranteed to converge, and the coordinatewise minimum point produced is statistically accurate under regularity conditions. Our theoretical investigations focus on nonasymptotic robust analysis, which demonstrates that joint rank reduction and outlier detection leads to improved prediction accuracy. In particular, we show that redescending psi-functions can essentially attain the minimax optimal error rate, and in some less challenging problems convex regularization guarantees the same low error rate. The performance of the proposed method is examined by simulation studies and real data examples.