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?