According to this paper by Meinshausen and Bühlmann (2006) the variable subset selection coming out of a Lasso is not always consistent in high-dimensional cases. It is bounded by the neighbourhood stability criterion. Is there somebody who can explain me in layman terms what this condition implies?

If I understand correctly, the constraint implies that Lasso can find a consistent subset of variables, even in high-dimensional cases, but only if the sparsity $S$ of datamatrix $X$ is not too much compared to the number of variables $p$.

  • $\begingroup$ The M&B paper is a seminal paper in the field, but not the easiest to read. Check out Chapters 10 & 11 of Statistical Learning with Sparsity (free at web.stanford.edu/~hastie/StatLearnSparsity_files/SLS.pdf) for a more pedagogical explanation of the necessary conditions for the lasso to correctly recover the support. The long and short is that the "true" variables cannot be too correlated with i) each other; and ii) the false variables. The sparsity of $X$ does not matter directly (though if $X$ is almost all zeros, there may be more correlation among its columns). $\endgroup$ – mweylandt May 22 '18 at 19:24

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