GENERALIZED MATRIX DECOMPOSITION REGRESSION: ESTIMATION AND INFERENCE FOR TWO-WAY STRUCTURED DATA.

Motivated by emerging applications in ecology, microbiology, and neuroscience, 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 auxiliary information on row and column structures. GMDR extends the principal component regression (PCR) to two-way structured data, but unlike PCR, 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 incorporating relevant auxiliary row and column structures. As a result, GMDI does not require the true regression coefficients to be sparse, but constrains the coordinate system representing the regression coefficients according to the column structure. GMDI 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 in simulation studies and an application to human microbiome data.