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I am currently using the lasso for variable selection. Based on random subsamples of my data, the selections differ a lot, i.e. the selection consistency is bad. Literature such as Meinshausen & Bühlmann (2006), Zhao & Yu (2006), and Zou (2006) describe a so called "irrepresentable condition", which needs to be fulfilled in order to provide consistent variable selection. However, literature about the topic is quite technical, leaving me behind with several questions.

Question 1: How could "irrepresentable condition" be explained in easy/non-technical words?

Question 2: Is it realistic to fulfill the "irrepresentable condition" in practice or is it just a theoretical concept?

Question 3: If the "irrepresentable condition" does not work in practice, are there other ways to provide consistent variable selection?

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    $\begingroup$ 2) It is a technical assumption involving the knowledge of the sparse set of regressors and as such may in practice not be validated correctly in general (however, see the examples in Zhao & Yu (2006) for some sufficient "conditions" on the correlation structures). 3) In their paper "On the conditions used to prove oracle results for the Lasso", van de Geer and Bühlmann (2009) show the relationship between different assumptions on the lasso needed to prove the main theorems in the lasso literature. $\endgroup$
    – chRrr
    Oct 11, 2017 at 13:28
  • $\begingroup$ Thanks a lot @chRrr for the explanation and the very helpful paper! $\endgroup$
    – Joachim Schork
    Oct 12, 2017 at 12:26

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