Consistency of the group Lasso and multiple kernel learning
Consistency of the group Lasso and multiple kernel learning
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DOI：
10.5555/1390681.1390721
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发表时间：
2007-07
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影响因子：
--
通讯作者：
F. Bach
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文献类型：
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作者：
F. Bach
我们考虑最小二乘回归问题与正规化的块l1-范数，即，欧几里德规范的总和在空间的维度大于1。这个问题，被称为群Lasso，通过l1范数扩展了通常的正则化，其中所有空间都是一维的，通常被称为Lasso。本文研究了群Lasso的渐近群选择相容性。在实际的假设条件下，如模型不规范，我们得到了群Lasso相容的充分必要条件。当线性预测和欧几里得范数被函数和再生核希尔伯特范数取代时，该问题通常被称为多核学习，通常用于从异构数据源学习和非线性变量选择。使用工具从功能分析，特别是covar算子，我们扩展到这个无限维的情况下的一致性结果，并提出了一个自适应计划，以获得一致的模型估计，即使当所需的非自适应计划的必要条件不满足。
We consider the least-square regression problem with regularization by a block l1-norm, that is, a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the l1-norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic group selection consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model mis specification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covar iance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.