Stability Approach to Regularization Selection for Reduced-Rank Regression

TL;DR

This paper introduces StARS-RRR, a stability-based method for selecting the rank in reduced-rank regression, demonstrating theoretical consistency and superior empirical performance over traditional methods.

Contribution

The paper proposes a novel stability approach, StARS-RRR, for rank estimation in reduced-rank regression, with proven consistency and improved accuracy over existing methods.

Findings

StARS-RRR achieves rank estimation consistency.

StARS-RRR outperforms AIC, BIC, and cross validation in simulations.

Applied to breast cancer data, StARS-RRR identifies relevant genetic pathways with lower prediction error.

Abstract

The reduced-rank regression model is a popular model to deal with multivariate response and multiple predictors, and is widely used in biology, chemometrics, econometrics, engineering, and other fields. In the reduced-rank regression modelling, a central objective is to estimate the rank of the coefficient matrix that represents the number of effective latent factors in predicting the multivariate response. Although theoretical results such as rank estimation consistency have been established for various methods, in practice rank determination still relies on information criterion based methods such as AIC and BIC or subsampling based methods such as cross validation. Unfortunately, the theoretical properties of these practical methods are largely unknown. In this paper, we present a novel method called StARS-RRR that selects the tuning parameter and then estimates the rank of the coefficient matrix for reduced-rank regression based on the stability approach. We prove that StARS-RRR achieves rank estimation consistency, i.e., the rank estimated with the tuning parameter selected by StARS-RRR is consistent to the true rank. Through a simulation study, we show that StARS-RRR outperforms other tuning parameter selection methods including AIC, BIC, and cross validation as it provides the most accurate estimated rank. In addition, when applied to a breast cancer dataset, StARS-RRR discovers a reasonable number of genetic pathways that affect the DNA copy number variations and results in a smaller prediction error than the other methods with a random-splitting process.