The PhaseLift for Non-quadratic Gaussian Measurements

We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional measurements of the form $y=f(\mathbf{a}^T\mathbf{x}_0)$ for some nonlinear function $f$. When the measurement vector $\mathbf a$ is iid Gaussian, Brillinger observed in his 1982 paper that $\mu_\ell\cdot\mathbf{x}_0 = \min_{\mathbf{x}}\mathbb{E}(y - \mathbf{a}^T\mathbf{x})^2$, where $\mu_\ell=\mathbb{E}_{\gamma}[\gamma f(\gamma)]$ with $\gamma$ 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 $\mathbf{x}_0$ when $\mu_\ell=0$. 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.