Restricted Eigenvalue and Irrepresentable Conditions

Lasso regression is popular precisely because it can select a handful of genuinely important variables out of thousands of candidates by shrinking irrelevant coefficients to exactly zero. But that selection is only trustworthy under specific conditions on how correlated the candidate predictors are with each other — with thousands of correlated factors (momentum, value, quality variants, sector dummies), those conditions can fail silently, and Lasso can quietly pick the wrong variables while still looking like it fit well.

The restricted eigenvalue condition asks: among vectors that are "sparse-like" (concentrated on a small number of coordinates, roughly matching the true underlying model), does the predictor matrix still behave like it has full rank — i.e., does it never map such a vector nearly to zero? If predictors are so collinear that some sparse combination of them is almost perfectly redundant with another, the condition fails, and Lasso's prediction accuracy guarantee (that it approximates the true regression function well) breaks down.

The irrepresentable condition is stricter and specifically about exact variable selection: it requires that the truly irrelevant predictors cannot be too well "represented" as a combination of the truly relevant ones. Concretely, if an irrelevant factor is a near-perfect linear combination of two relevant factors, the irrepresentable condition fails, and Lasso can end up selecting the irrelevant factor instead of (or alongside) the real ones, even with an enormous amount of data — no amount of additional history fixes a structural correlation problem in the predictors themselves.