Abstract:
It has been shown recently that the Lasso can be sign consistent for linear regression -- picking exactly the right variables -- only under a restrictive condition on the design matrix. I will give some geometric illustration of these results. Even if the condition is not fulfilled, it can be shown that the Lasso estimator is close to the true vector of regression coefficients under much more general assumptions in high-dimensional problems, meaning that falsely selected variables have asymptotically only very small coefficients. Most of the results assume that the true regression vector is sparse in the sense that only a few entries are non-zero. An extension to weak l_p-balls with p < 1 can also be obtained. The results will be illustrated by an example of frequency detection in the search for variable stars.