Title: Efficient parametric approximations of neural net function space distance. Abstract: It is often useful to compactly summarize important properties of a training dataset so that they can be used later without storing and/or iterating over the entire dataset. We consider a specific case of this: approximating the function space distance (FSD) over the training set, i.e. the average distance between the outputs of two neural networks. We propose an efficient approximation to FSD for ReLU neural networks based on approximating the architecture as a linear network with stochastic gating. Despite requiring only one parameter per unit of the network, our approach outcompetes other parametric approximations with larger memory requirements. Applied to continual learning, our parametric approximation is competitive with state-of-the-art nonparametric approximations which require storing many training examples. Furthermore, we show its efficacy in influence function estimation, allowing influence functions to be accurately estimated without iterating over the full dataset.