Efficient Algorithms for Non-gaussian Single Index Models with Generative Priors

In this work, we focus on high-dimensional single index models with non-Gaussian sensing vectors and generative priors. More specifically, our goal is to estimate the underlying signal from i.i.d. realizations of the semi-parameterized single index model, where the underlying signal is contained in (up to a constant scaling) the range of a Lipschitz continuous generative model with bounded low-dimensional inputs, the sensing vector follows a non-Gaussian distribution, the noise is a random variable that is independent of the sensing vector, and the unknown non-linear link function is differentiable. Using the first- and second-order Stein's identity, we introduce efficient algorithms to obtain estimated vectors that achieve the near-optimal statistical rate. Experimental results on image datasets are provided to support our theory.