Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions

Learning distance functions between complex objects, such as the Wasserstein distance to compare point sets, is a common goal in machine learning applications. However, functions on such complex objects (e.g., point sets and graphs) are often required to be invariant to a wide variety of group actions e.g. Therefore, continuous and symmetric *product* functions (such as distance functions) on such complex objects must also be invariant to the *product* of such group actions. We call these functions symmetric and factor-wise group invariant functions (or SGFI functions} in short).In this paper, we first present a general neural network architecture for approximating SFGI functions. The main contribution of this paper combines this general NN with a sketching idea in order to develop a specific and efficient neural network which can approximate the p -th Wasserstein distance between point sets.Very importantly, the required model complexity is *independent* of the sizes of input point sets.