New Perspectives on k-Support and Cluster Norms
New Perspectives on k-Support and Cluster Norms
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发表时间：
2014-03
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通讯作者：
Andrew M. McDonald;M. Pontil;Dimitris Stamos
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作者：
Andrew M. McDonald;M. Pontil;Dimitris Stamos
我们研究了一个正则化器，它被定义为二次方程的参数化下确界，我们称之为盒范数。我们证明了 k-support 范数，一种由 [Argyriou et al, 2012] 提出的用于稀疏向量预测问题的正则化器，属于这个家族，并且 box-norm 可以作为前者的扰动生成。我们推导了一种改进的算法来计算平方框范数的邻近算子，并且我们提供了一种计算范数的方法。我们将范数扩展到矩阵，引入谱 k-支持范数和谱盒范数。我们注意到，谱盒范数本质上等同于簇范数，这是由 [Jacob 等人引入的多任务学习正则化器。 2009a]，这又可以解释为谱 k 支持范数的扰动。将范数居中对于多任务学习很重要，我们还提供了一种使用范数的中心版本作为正则化器的方法。数值实验表明，谱 k 支持和框范数及其中心变体分别在矩阵补全和多任务学习问题中提供了最先进的性能。
We study a regularizer which is defined as a parameterized infimum of quadratics, and which we call the box-norm. We show that the k-support norm, a regularizer proposed by [Argyriou et al, 2012] for sparse vector prediction problems, belongs to this family, and the box-norm can be generated as a perturbation of the former. We derive an improved algorithm to compute the proximity operator of the squared box-norm, and we provide a method to compute the norm. We extend the norms to matrices, introducing the spectral k-support norm and spectral box-norm. We note that the spectral box-norm is essentially equivalent to the cluster norm, a multitask learning regularizer introduced by [Jacob et al. 2009a], and which in turn can be interpreted as a perturbation of the spectral k-support norm. Centering the norm is important for multitask learning and we also provide a method to use centered versions of the norms as regularizers. Numerical experiments indicate that the spectral k-support and box-norms and their centered variants provide state of the art performance in matrix completion and multitask learning problems respectively.