This paper explores the asymptotic distributions of low-dimensional patterns in linear regression with regularizers such as Lasso, Elastic Net, Generalized Lasso, and SLOPE, as the number of observations n grows and the penalty increases at rate \(\sqrt{n}\) . While the asymptotic distribution of rescaled estimation errors is well-understood, convergence of patterns lacks proof in the literature, even for Lasso. We provide a proof using the Hausdorff distance for subdifferentials. We also derive the limiting probability of recovering the true model pattern, which approaches 1 when the penalty scaling diverges and the regularizer-specific asymptotic irrepresentability condition is satisfied. We propose two-step procedures that asymptotically recover model patterns, regardless of the irrepresentability condition. Our theory shows that Fused Lasso cannot reliably recover its clustering pattern for independent regressors, but this can be resolved by concavifying its penalty coefficients. Simulation studies compare the asymptotic properties of Lasso, Fused Lasso, and SLOPE.
