# Lifting high-dimensional nonlinear models with Gaussian regressors

arXiv: https://arxiv.org/abs/1712.03638 · HTML: https://mlsanity.com/paper/1712.03638/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2017.

## Built on by (most cited first, 1 of 1)

- [Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2308.14507/) (2023)

## Abstract (arXiv)

We study the problem of recovering a structured signal x0 from high-dimensional data yi=f(aiTx0) for some nonlinear (and potentially unknown) link function f, when the regressors ai are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates x0 up to a constant of proportionality μℓ, 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 x0 when μℓ=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.