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It is known that for a certain class of single index models (SIMs) $Y =f(\boldsymbol{X}_{p \times 1}^\intercal\boldsymbol{\beta}_0, \varepsilon)$,support recovery is impossible when $\boldsymbol{X} \sim \mathcal{N}(0,\mathbb{I}_{p \times p})$ and a model complexity adjusted sample size is belowa critical threshold. Recently, optimal algorithms based on Sliced InverseRegression (SIR) were suggested. These algorithms work provably under theassumption that the design $\boldsymbol{X}$ comes from an i.i.d. Gaussiandistribution. In the present paper we analyze algorithms based on covariancescreening and least squares with $L_1$ penalization (i.e. LASSO) anddemonstrate that they can also enjoy optimal (up to a scalar) rescaled samplesize in terms of support recovery, albeit under slightly different assumptionson $f$ and $\varepsilon$ compared to the SIR based algorithms. Furthermore, weshow more generally, that LASSO succeeds in recovering the signed support of$\boldsymbol{\beta}_0$ if $\boldsymbol{X} \sim \mathcal{N}(0,\boldsymbol{\Sigma})$, and the covariance $\boldsymbol{\Sigma}$ satisfies theirrepresentable condition. Our work extends existing results on the supportrecovery of LASSO for the linear model, to a more general class of SIMs.

L1-Regularized Least Squares for Support Recovery of High Dimensional Single Index Models with Gaussian Designs