Convex Relaxation for Combinatorial Penalties

Guillaume Obozinski; Francis Bach

hal-00694765, version 1

Convex Relaxation for Combinatorial Penalties

Guillaume Obozinski (, ) ^1, 2, Francis Bach (, ) ^1, 2

Abstract: In this paper, we propose an unifying view of several recently proposed structured sparsity-inducing norms. We consider the situation of a model simultaneously (a) penalized by a set- function de ned on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in Lp-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.

1: SIERRA (INRIA Paris - Rocquencourt)
INRIA : PARIS - ROCQUENCOURT – Ecole normale supérieure de Paris - ENS Paris – CNRS : UMR8548
2: Laboratoire d'informatique de l'école normale supérieure (LIENS)
CNRS : UMR8548 – Ecole normale supérieure de Paris - ENS Paris

hal-00694765, version 1
http://hal.archives-ouvertes.fr/hal-00694765
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From: Guillaume Obozinski <>
Submitted on: Sunday, 6 May 2012 15:44:49
Updated on: Sunday, 6 May 2012 21:54:36

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arXiv: 1205.1240

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