  In this paper, we study the problem of signal estimation from noisy
non-linear measurements when the unknown $n$-dimensional signal is in the range
of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional
inputs. We make the assumption of sub-Gaussian measurements, which is satisfied
by a wide range of measurement models, such as linear, logistic, 1-bit, and
other quantized models. In addition, we consider the impact of adversarial
corruptions on these measurements. Our analysis is based on a generalized Lasso
approach (Plan and Vershynin, 2016). We first provide a non-uniform recovery
guarantee, 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 adversarial
noise. Then, we apply this result to neural network generative models, and
discuss various extensions to other models and non-i.i.d.~measurements.
Moreover, we show that our result can be extended to the uniform recovery
guarantee under the assumption of a so-called local embedding property, which
is satisfied by the 1-bit and censored Tobit models.
