Model Selection for High-Dimensional Regression under the Generalized Irrepresentability Condition
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In the high-dimensional regression model a response variable is linearly related to p covariates, but the sample size n is smaller than p. We assume that only a small subset of covariates is `active' (i.e., the corresponding coefficients are non-zero), and consider the model-selection problem of identifying the active covariates. A popular approach is to estimate the regression coefficients through the Lasso (ℓ_1-regularized least squares). This is known to correctly identify the active set only if the irrelevant covariates are roughly orthogonal to the relevant ones, as quantified through the so called `irrepresentability' condition. In this paper we study the `Gauss-Lasso' selector, a simple two-stage method that first solves the Lasso, and then performs ordinary least squares restricted to the Lasso active set. We formulate `generalized irrepresentability condition' (GIC), an assumption that is substantially weaker than irrepresentability. We prove that, under GIC, the Gauss-Lasso correctly recovers the active set.
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