Fast Approximation of the Generalized Sliced-Wasserstein Distance

Generalized sliced-Wasserstein distance is a variant of slicedWasserstein 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 slicedWasserstein 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.