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

Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priors