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Publication Detail
Incorporating additional constraints in sparse estimation
Abstract
It is well known that a linear regression can benefit from knowledge that the underlying regression vector is sparse. The combinatorial problem of selecting the nonzero components of this vector can be relaxed by regularizing the squared error with a convex penalty function like the ℓ 1 norm. However, in many applications, additional conditions on the structure of the regression vector and its sparsity pattern are available. Incorporating this information into the learning method may lead to a significant decrease of the estimation error. In this paper, we review a recently proposed family of convex penalty functions, which encode prior knowledge on the structure of the vector formed by the absolute values of the regression coefficients. This family subsumes the ℓ 1 norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We discuss special cases of these penalty functions in which the regularized empirical error function can be efficiently minimized by a proximal-point method. We compare this method to a previous method based on block coordinate descent and present numerical experiments which highlight the benefit of our framework over a greedy approach and the Lasso method. © 2012 IFAC.
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