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We consider the homogeneous and the non-homogeneous convex relaxations forcombinatorial penalty functions defined on support sets. Our study identifieskey differences in the tightness of the resulting relaxations through thenotion of the lower combinatorial envelope of a set-function along with newnecessary conditions for support identification. We then propose a generaladaptive estimator for convex monotone regularizers, and derive new sufficientconditions for support recovery in the asymptotic setting.

Combinatorial Penalties: Which structures are preserved by convex relaxations?