Robust Reduced Rank Regression

TL;DR

This paper introduces a robust reduced-rank regression method that effectively handles outliers in high-dimensional multivariate data, improving estimation accuracy and enabling outlier detection through a novel regularized approach.

Contribution

It proposes a new robust reduced-rank regression framework with an efficient algorithm, unifying and extending existing robust multivariate methods, with proven convergence and statistical accuracy.

Findings

The method achieves minimax optimal error rates with redescending ψ-functions.

Convex regularization guarantees low error in less challenging scenarios.

Simulation and real data demonstrate improved robustness and accuracy.

Abstract

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.