  We consider high-dimensional regression over subgroups of observations. Our
work is motivated by biomedical problems, where disease subtypes, for example,
may differ with respect to underlying regression models, but sample sizes at
the subgroup-level may be limited. We focus on the case in which
subgroup-specific models may be expected to be similar but not necessarily
identical. Our approach is to treat subgroups as related problem instances and
jointly estimate subgroup-specific regression coefficients. This is done in a
penalized framework, combining an $\ell_1$ term with an additional term that
penalizes differences between subgroup-specific coefficients. This gives
solutions that are globally sparse but that allow information-sharing between
the subgroups. We present algorithms for estimation and empirical results on
simulated data and using Alzheimer's disease, amyotrophic lateral sclerosis and
cancer datasets. These examples demonstrate the gains our approach can offer in
terms of prediction and the ability to estimate subgroup-specific sparsity
patterns.
