  In this paper, we aim to estimate the direction of an underlying signal from
its nonlinear observations following the semi-parametric single index model
(SIM). Unlike conventional compressed sensing where the signal is assumed to be
sparse, we assume that the signal lies in the range of an $L$-Lipschitz
continuous generative model with bounded $k$-dimensional inputs. This is mainly
motivated by the tremendous success of deep generative models in various real
applications. Our reconstruction method is non-iterative (though approximating
the projection step may use an iterative procedure) and highly efficient, and
it is shown to attain the near-optimal statistical rate of order $\sqrt{(k \log
L)/m}$, where $m$ is the number of measurements. We consider two specific
instances of the SIM, namely noisy $1$-bit and cubic measurement models, and
perform experiments on image datasets to demonstrate the efficacy of our
method. In particular, for the noisy $1$-bit measurement model, we show that
our non-iterative method significantly outperforms a state-of-the-art iterative
method in terms of both accuracy and efficiency.
