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

arXiv: https://arxiv.org/abs/1511.08102 · HTML: https://mlsanity.com/paper/1511.08102/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2015.

## Built on by (most cited first, 6 of 6)

- [Non-Iterative Recovery from Nonlinear Observations using Generative Models](https://mlsanity.com/paper/2205.15749/) (2022)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [Single Index Bandits: Generalized Linear Contextual Bandits with Unknown Reward Functions](https://mlsanity.com/paper/2506.12751/) (2025)
- [Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension](https://mlsanity.com/paper/2111.14486/) (2021)
- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [A Consistent and Scalable Algorithm for Best Subset Selection in Single Index Models](https://mlsanity.com/paper/2309.06230/) (2023)

## Abstract (arXiv)

It is known that for a certain class of single index models (SIMs) Y = f(boldsymbolXp × 1^intercalboldsymbolβ0, varepsilon), support recovery is impossible when boldsymbolX ~ N(0, mathbbIp × 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 boldsymbolX comes from an i.i.d. Gaussian distribution. In the present paper we analyze algorithms based on covariance screening and least squares with L1 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 varepsilon compared to the SIR based algorithms. Furthermore, we show more generally, that LASSO succeeds in recovering the signed support of boldsymbolβ0 if boldsymbolX ~ N(0, boldsymbolSigma), and the covariance boldsymbolSigma 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.