  This paper explores the homogeneity of coefficients in high-dimensional
regression, which extends the sparsity concept and is more general and suitable
for many applications. Homogeneity arises when one expects regression
coefficients corresponding to neighboring geographical regions or a similar
cluster of covariates to be approximately the same. Sparsity corresponds to a
special case of homogeneity with a known atom zero. In this article, we propose
a new method called clustering algorithm in regression via data-driven
segmentation (CARDS) to explore homogeneity. New mathematics are provided on
the gain that can be achieved by exploring homogeneity. Statistical properties
of two versions of CARDS are analyzed. In particular, the asymptotic normality
of our proposed CARDS estimator is established, which reveals better estimation
accuracy for homogeneous parameters than that without homogeneity exploration.
When our methods are combined with sparsity exploration, further efficiency can
be achieved beyond the exploration of sparsity alone. This provides additional
insights into the power of exploring low-dimensional strucuture in
high-dimensional regression: homogeneity and sparsity. The newly developed
method is further illustrated by simulation studies and applications to real
data.
