I've grasped from a paper (https://www.stat.cmu.edu/%7Eryantibs/papers/lassounique.pdf) that Lasso may not yield a unique solution when the number of variables (p) exceeds the number of observations (n). Given that the oracle property in variable selection hinges on a method providing a unique solution (as far as I understood), does Lasso fail to exhibit this property under certain conditions?

If this assertion holds true, how can I empirically determine whether a Lasso model over a given dataset produces a unique solution or not? Are there theoretical underpinnings to assess uniqueness in applied scenarios, or do we rely on simulations to demonstrate uniqueness or non-uniqueness in specific cases?


Based on the article (https://www.stat.cmu.edu/~ryantibs/papers/lassounique.pdf) and on this post What causes lasso to be unstable for feature selection? made by Xavier Bourret Sicotte, Lasso needs to fullfil the following to offer an unique solution:

  1. Linearly independent

  2. Affinely independent When the columns Xs are in general position.

  3. When the columns X are from a continuous distribution

The question remains how this should be shown in an applied work?

  • $\begingroup$ See What causes lasso to be unstable for feature selection? or The lasso problem and uniqueness. $\endgroup$ Commented Jun 11 at 9:44
  • $\begingroup$ @user2974951, your second link seems to point to the same paper (though a different version) as the OP's link. $\endgroup$ Commented Jun 11 at 10:08
  • $\begingroup$ If I remember correctly, adaptive lasso has the oracle property, but simple lasso does not. $\endgroup$ Commented Jun 11 at 10:09
  • $\begingroup$ In addition to the link, consider including a full reference to the paper. Links tend to go dead over time... $\endgroup$ Commented 2 days ago


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