  In this paper, we propose projected gradient descent (PGD) algorithms for
signal estimation from noisy nonlinear measurements. We assume that the unknown
$p$-dimensional signal lies near the range of an $L$-Lipschitz continuous
generative model with bounded $k$-dimensional inputs. In particular, we
consider two cases when the nonlinear link function is either unknown or known.
For unknown nonlinearity, similarly to \cite{liu2020generalized}, we make the
assumption of sub-Gaussian observations and propose a linear least-squares
estimator. We show that when there is no representation error and the sensing
vectors are Gaussian, roughly $O(k \log L)$ samples suffice to ensure that a
PGD algorithm converges linearly to a point achieving the optimal statistical
rate using arbitrary initialization. For known nonlinearity, we assume
monotonicity as in \cite{yang2016sparse}, and make much weaker assumptions on
the sensing vectors and allow for representation error. We propose a nonlinear
least-squares estimator that is guaranteed to enjoy an optimal statistical
rate. A corresponding PGD algorithm is provided and is shown to also converge
linearly to the estimator using arbitrary initialization. In addition, we
present experimental results on image datasets to demonstrate the performance
of our PGD algorithms.
