Fast Approximation of the Generalized Sliced-Wasserstein Distance

Generalized sliced-Wasserstein distance is a variant of sliced-Wasserstein distance that exploits the power of non-linear projection through a given defining function to better capture the complex structures of probability distributions. Similar to the sliced-Wasserstein distance, generalized sliced-Wasserstein is defined as an expectation over random projections which can be approximated by the Monte Carlo method. However, the complexity of that approximation can be expensive in high-dimensional settings. To that end, we propose to form deterministic and fast approximations of the generalized sliced-Wasserstein distance by using the concentration of random projections when the defining functions are polynomial function and neural network type function. Our approximations hinge upon an important result that one-dimensional projections of a high-dimensional random vector are approximately Gaussian.