  We consider the least-square linear regression problem with regularization by
the $\ell^1$-norm, a problem usually referred to as the Lasso. In this paper,
we first present a detailed asymptotic analysis of model consistency of the
Lasso in low-dimensional settings. For various decays of the regularization
parameter, we compute asymptotic equivalents of the probability of correct
model selection. For a specific rate decay, we show that the Lasso selects all
the variables that should enter the model with probability tending to one
exponentially fast, while it selects all other variables with strictly positive
probability. We show that this property implies that if we run the Lasso for
several bootstrapped replications of a given sample, then intersecting the
supports of the Lasso bootstrap estimates leads to consistent model selection.
This novel variable selection procedure, referred to as the Bolasso, is
extended to high-dimensional settings by a provably consistent two-step
procedure.
