Generalized Matrix Decomposition Regression: Estimation and Inference for Two-way Structured Data

TL;DR

This paper introduces GMDR and GMDI, novel methods for high-dimensional regression with two-way structured data, improving prediction accuracy and inference flexibility by leveraging auxiliary information and addressing heteroscedasticity.

Contribution

The paper develops GMDR and GMDI, extending principal component regression and inference to two-way structured data with enhanced prediction and inference capabilities.

Findings

GMDR outperforms traditional methods in prediction accuracy.

GMDI maintains valid inference without requiring sparsity.

Methods are effective in microbiome data analysis.

Abstract

This paper studies high-dimensional regression with two-way structured data. To estimate the high-dimensional coefficient vector, we propose the generalized matrix decomposition regression (GMDR) to efficiently leverage any auxiliary information on row and column structures. The GMDR extends the principal component regression (PCR) to two-way structured data, but unlike PCR, the GMDR selects the components that are most predictive of the outcome, leading to more accurate prediction. For inference on regression coefficients of individual variables, we propose the generalized matrix decomposition inference (GMDI), a general high-dimensional inferential framework for a large family of estimators that include the proposed GMDR estimator. GMDI provides more flexibility for modeling relevant auxiliary row and column structures. As a result, GMDI does not require the true regression coefficients to be sparse; it also allows dependent and heteroscedastic observations. We study the theoretical properties of GMDI in terms of both the type-I error rate and power and demonstrate the effectiveness of GMDR and GMDI on simulation studies and an application to human microbiome data.