  The estimation problem in a high regression model with structured sparsity is
investigated. An algorithm using a two steps block thresholding procedure
called GR-LOL is provided. Convergence rates are produced: they depend on
simple coherence-type indices of the Gram matrix -easily checkable on the data-
as well as sparsity assumptions of the model parameters measured by a
combination of $l_1$ within-blocks with $l_q,q<1$ between-blocks norms. The
simplicity of the coherence indicator suggests ways to optimize the rates of
convergence when the group structure is not naturally given by the problem and
is unknown. In such a case, an auto-driven procedure is provided to determine
the regressors groups (number and contents). An intensive practical study
compares our grouping methods with the standard LOL algorithm. We prove that
the grouping rarely deteriorates the results but can improve them very
significantly. GR-LOL is also compared with group-Lasso procedures and exhibits
a very encouraging behavior. The results are quite impressive, especially when
GR-LOL algorithm is combined with a grouping pre-processing.
