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.

本文提出了一种非线性观测下基于单指数模型和连续型生成模型的信号方向估计方法，具有高效、不迭代的优点，并且在测试中表现优于迭代方法。