,
Consistency of the group Lasso and multiple kernel learning
(2007)cite arxiv:0707.3390.
Аннотация
We consider the least-square regression problem with regularization by a
block 1-norm, i.e., 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 1-norm where all spaces have dimension one, where it is
commonly referred to as the Lasso. In this paper, we study the asymptotic model
consistency of the group Lasso. We derive necessary and sufficient conditions
for the consistency of group Lasso under practical assumptions, such as model
misspecification. 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 covariance 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.
