---
abstract: 'We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\y_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\ab_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\x_0$ up to a constant of proportionality $\mu_\ell$, which depends on $f$. Recently, Plan & Vershynin (2015) extended this result to the high-dimensional setting deriving sharp error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $\mu_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an alternative convex recovery method. In a nutshell, our method treats such link functions as if they were linear in a lifted space of higher-dimension. Interestingly, our error analysis captures the effect of both the nonlinearity and the problem’s geometry in a few simple summary parameters.'
author:
- Christos Thrampoulidis
- Ankit Singh Rawat
bibliography:
- 'compbib.bib'
title: |
     Lifting high-dimensional nonlinear models\
    with Gaussian regressors 
---


