Lifting high-dimensional nonlinear models with Gaussian regressors [article]

Christos Thrampoulidis, Ankit Singh Rawat
2018 arXiv   pre-print
We study the problem of recovering a structured signal x_0 from high-dimensional data y_i=f(a_i^Tx_0) for some nonlinear (and potentially unknown) link function f, when the regressors a_i are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates x_0 up to a constant of proportionality μ_ℓ, 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
more » ... t-squares and the Lasso fail to recover x_0 when μ_ℓ=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.
arXiv:1712.03638v2 fatcat:62lvexbjyvggtbiyb2laotme7i