Convex Relaxation for Combinatorial Penalties

In this paper, we propose an unifying view of several recently proposed structured sparsityinducing norms. We consider the situation of a model simultaneously (a) penalized by a setfunction defined on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in `p-norm. We show that the natural combinatorial optimization problems obtained may be relaxed into convex optimization problems and introduce a notion, the lower combinatorial envelope of a set-function, that characterizes the tightness of our relaxations. We moreover establish links with norms based on latent representations including the latent group Lasso and block-coding, and with norms obtained from submodular functions.