The PhaseLift for Non-quadratic Gaussian Measurements
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We study the problem of recovering a structured signal x_0 from high-dimensional measurements of the form y=f(a^Tx_0) for some nonlinear function f. When the measurement vector a is iid Gaussian, Brillinger observed in his 1982 paper that μ_ℓ·x_0 = _xE(y - a^Tx)^2, where μ_ℓ=E_γ[γ f(γ)] with γ being a standard Gaussian random variable. Based on this simple observation, he showed that, in the classical statistical setting, the least-squares method is consistent. More recently, Plan & Vershynin extended this result to the high-dimensional setting and derived error bounds for the generalized Lasso. Unfortunately, both least-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 appropriate generic semidefinite-optimization based method. In a nutshell, our idea is to treat such link functions as if they were linear in a lifted space of higher-dimension. An appealing feature of our error analysis is that it captures the effect of the nonlinearity in a few simple summary parameters, which can be particularly useful in system design.
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