Title: Supervised Random Feature Regression via Projection Pursuit. Abstract: Random feature methods and neural network models are two popular nonparametric modeling methods, which are regarded as representatives of shallow learning and Neural Network, respectively. In practice random  feature methods are short of the capacity of feature learning, while neural network methods lead to computationally heavy problems. This paper aims at proposing  a flexible but computational efficient method for general nonparametric problems.  Precisely, our proposed method is a feed-forward two-layer nonparametric estimation, and the first layer is used to learn a series of univariate basis functions for each projection variable, and then search for their optimal linear combination for each group of these learnt functions. Based on all the  features derived in the first layer, the second layer attempts at learning a single index function with an unknown activation function. Our nonparametric estimation takes advantage of both random features and neural networks, and can be seen as an intermediate bridge between them.