  An approximate method for conducting resampling in Lasso, the $\ell_1$
penalized linear regression, in a semi-analytic manner is developed, whereby
the average over the resampled datasets is directly computed without repeated
numerical sampling, thus enabling an inference free of the statistical
fluctuations due to sampling finiteness, as well as a significant reduction of
computational time. The proposed method is based on a message passing type
algorithm, and its fast convergence is guaranteed by the state evolution
analysis, when covariates are provided as zero-mean independently and
identically distributed Gaussian random variables. It is employed to implement
bootstrapped Lasso (Bolasso) and stability selection, both of which are
variable selection methods using resampling in conjunction with Lasso, and
resolves their disadvantage regarding computational cost. To examine
approximation accuracy and efficiency, numerical experiments were carried out
using simulated datasets. Moreover, an application to a real-world dataset, the
wine quality dataset, is presented. To process such real-world datasets, an
objective criterion for determining the relevance of selected variables is also
introduced by the addition of noise variables and resampling.
