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In this paper, we study the problem of signal estimation from noisynon-linear measurements when the unknown $n$-dimensional signal is in the rangeof an $L$-Lipschitz continuous generative model with bounded $k$-dimensionalinputs. We make the assumption of sub-Gaussian measurements, which is satisfiedby a wide range of measurement models, such as linear, logistic, 1-bit, andother quantized models. In addition, we consider the impact of adversarialcorruptions on these measurements. Our analysis is based on a generalized Lassoapproach (Plan and Vershynin, 2016). We first provide a non-uniform recoveryguarantee, which states that under i.i.d.~Gaussian measurements, roughly$O\left(\frac{k}{\epsilon^2}\log L\right)$ samples suffice for recovery with an$\ell_2$-error of $\epsilon$, and that this scheme is robust to adversarialnoise. Then, we apply this result to neural network generative models, anddiscuss various extensions to other models and non-i.i.d.~measurements.Moreover, we show that our result can be extended to the uniform recoveryguarantee under the assumption of a so-called local embedding property, whichis satisfied by the 1-bit and censored Tobit models.

The Generalized Lasso with Nonlinear Observations and Generative Priors