## Consistency of the Group Lasso and Multiple Kernel Learning

** Francis R. Bach**; 9(40):1179−1225, 2008.

### Abstract

We consider the least-square regression problem with regularization by
a block *l*_{1}-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 *l*_{1}-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.

© JMLR 2008. (edit, beta) |