  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.
