Generalized Reduced-Rank Regression with Homogeneity Pursuit

TL;DR

This paper introduces a unified framework for multi-response regression models that incorporates homogeneity, low rank, and sparsity assumptions, providing theoretical guarantees and demonstrating practical effectiveness.

Contribution

It develops a regularized maximum likelihood estimation method for high-dimensional models that unifies homogeneity, low rank, and sparsity assumptions, with proven statistical consistency.

Findings

The estimator is statistically consistent under mild conditions.

Homogeneity can significantly improve estimation accuracy in certain scenarios.

Numerical simulations and real data analysis validate the proposed method.

Abstract

Homogeneity, low rank, and sparsity are three widely adopted assumptions in multi-response regression models to address the curse of dimensionality and improve estimation accuracy. However, there is limited literature that examines these assumptions within a unified framework. In this paper, we investigate the homogeneity, low rank, and sparsity assumptions under the generalized linear model with high-dimensional responses and covariates, encompassing a wide range of practical applications. Our work establishes a comprehensive benchmark for comparing the effects of these three assumptions and introduces a regularized maximum likelihood estimation method to fit the corresponding models. Under mild conditions,we prove the statistical consistency of our estimator. Theoretical results provide insights into the role of homogeneity and offer a quantitative analysis of scenarios where homogeneity improves estimation accuracy. The proposed method's effectiveness is demonstrated through numerical simulations and an empirical analysis of tree species data from Barro Colorado Island.