Fast Approximation of the Generalized Sliced-Wasserstein Distance
This work addresses a computational bottleneck for researchers and practitioners using optimal transport metrics in machine learning, though it is incremental as it builds on existing generalized sliced-Wasserstein methods.
The paper tackles the computational expense of approximating the generalized sliced-Wasserstein distance in high-dimensional settings by proposing deterministic and fast approximations based on the concentration of random projections, achieving efficient computation for specific defining functions like polynomial, circular, and neural network types.
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 the probability distributions. Similar to 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, circular 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.