  Sorted L-One Penalized Estimation is a relatively new convex optimization
procedure which allows for adaptive selection of regressors under sparse high
dimensional designs. Here we extend the idea of SLOPE to deal with the
situation when one aims at selecting whole groups of explanatory variables
instead of single regressors. This approach is particularly useful when
variables in the same group are strongly correlated and thus true predictors
are difficult to distinguish from their correlated "neighbors"'. We formulate
the respective convex optimization problem, gSLOPE (group SLOPE), and propose
an efficient algorithm for its solution. We also define a notion of the group
false discovery rate (gFDR) and provide a choice of the sequence of tuning
parameters for gSLOPE so that gFDR is provably controlled at a prespecified
level if the groups of variables are orthogonal to each other. Moreover, we
prove that the resulting procedure adapts to unknown sparsity and is
asymptotically minimax with respect to the estimation of the proportions of
variance of the response variable explained by regressors from different
groups. We also provide a method for the choice of the regularizing sequence
when variables in different groups are not orthogonal but statistically
independent and illustrate its good properties with computer simulations.
