Non-Iterative Recovery From Nonlinear Observations Using Generative Models

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 for 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 require 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.