L1-Regularized Least Squares for Support Recovery of High Dimensional Single Index Models with Gaussian Designs
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It is known that for a certain class of single index models (SIMs) Y = f(X_p × 1^β_0, ε), support recovery is impossible when X∼N(0, I_p × p) and a model complexity adjusted sample size is below a critical threshold. Recently, optimal algorithms based on Sliced Inverse Regression (SIR) were suggested. These algorithms work provably under the assumption that the design X comes from an i.i.d. Gaussian distribution. In the present paper we analyze algorithms based on covariance screening and least squares with L_1 penalization (i.e. LASSO) and demonstrate that they can also enjoy optimal (up to a scalar) rescaled sample size in terms of support recovery, albeit under slightly different assumptions on f and ε compared to the SIR based algorithms. Furthermore, we show more generally, that LASSO succeeds in recovering the signed support of β_0 if X∼N(0, Σ), and the covariance Σ satisfies the irrepresentable condition. Our work extends existing results on the support recovery of LASSO for the linear model, to a more general class of SIMs.
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