  Motivated by diagnosing the COVID-19 disease using 2D image biomarkers from
computed tomography (CT) scans, we propose a novel latent matrix-factor
regression model to predict responses that may come from an exponential
distribution family, where covariates include high-dimensional matrix-variate
biomarkers. A latent generalized matrix regression (LaGMaR) is formulated,
where the latent predictor is a low-dimensional matrix factor score extracted
from the low-rank signal of the matrix variate through a cutting-edge matrix
factor model. Unlike the general spirit of penalizing vectorization plus the
necessity of tuning parameters in the literature, instead, our prediction
modeling in LaGMaR conducts dimension reduction that respects the geometry
characteristic of intrinsic two-dimensional structure of the matrix covariate
and thus avoids iteration. This greatly relieves the computation burden, and
meanwhile maintains structural information so that the latent matrix factor
feature can perfectly replace the intractable matrix-variate owing to
high-dimensionality. The estimation procedure of LaGMaR is subtly derived by
transforming the bilinear form matrix factor model onto a high-dimensional
vector factor model, so that the method of principle components can be applied.
We establish bilinear-form consistency of the estimated matrix coefficient of
the latent predictor and consistency of prediction. The proposed approach can
be implemented conveniently. Through simulation experiments, prediction
capability of LaGMaR is shown to outperform existing penalized methods under
diverse scenarios of generalized matrix regressions. Through the application to
a real COVID-19 dataset, the proposed approach is shown to predict efficiently
the COVID-19.
