Lifting high-dimensional nonlinear models with Gaussian regressors

TL;DR

This paper introduces a novel convex method for recovering high-dimensional nonlinear models with Gaussian regressors, effectively handling cases where traditional methods fail, such as even link functions, by lifting the problem into a higher-dimensional space.

Contribution

The paper proposes a new convex recovery approach that addresses limitations of least-squares and Lasso in nonlinear high-dimensional models, especially for even link functions.

Findings

The method successfully recovers signals where traditional methods fail.

Error bounds incorporate nonlinearity and geometry effects.

The approach extends recovery capabilities to a broader class of nonlinear models.

Abstract

We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\mathbf{y}_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\mathbf{a}_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\mathbf{x}_0$ 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.