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.