  In high dimensional regression, feature clustering by their effects on
outcomes is often as important as feature selection. For that purpose,
clustered Lasso and octagonal shrinkage and clustering algorithm for regression
(OSCAR) are used to make feature groups automatically by pairwise $L_1$ norm
and pairwise $L_\infty$ norm, respectively. This paper proposes efficient path
algorithms for clustered Lasso and OSCAR to construct solution paths with
respect to their regularization parameters. Despite too many terms in
exhaustive pairwise regularization, their computational costs are reduced by
using symmetry of those terms. Simple equivalent conditions to check
subgradient equations in each feature group are derived by some graph theories.
The proposed algorithms are shown to be more efficient than existing algorithms
in numerical experiments.
