  We propose a new approach to safe variable preselection in high-dimensional
penalized regression, such as the lasso. Preselection - to start with a
manageable set of covariates - has often been implemented without clear
appreciation of its potential bias. Based on sequential implementation of the
lasso with increasing lists of predictors, we find a new property of the set of
corresponding cross-validation curves, a pattern that we call freezing. It
allows to determine a subset of covariates with which we reach the same lasso
solution as would be obtained using the full set of covariates. Freezing has
not been characterized before and is different from recently discussed safe
rules for discarding predictors. We demonstrate by simulation that ranking
predictors by their univariate correlation with the outcome, leads in a
majority of cases to early freezing, giving a safe and efficient way of
focusing the lasso analysis on a smaller and manageable number of predictors.
We illustrate the applicability of our strategy in the context of a GWAS
analysis and on microarray genomic data. Freezing offers great potential for
extending the applicability of penalized regressions to ultra highdimensional
data sets. Its applicability is not limited to the standard lasso but is a
generic property of many penalized approaches.
