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 noniterative (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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{(k\log L)}/m$</tex> , 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.