Towards Better Spherical Sliced-Wasserstein Distance Learning with Data-Adaptive Discriminative Projection Direction

Spherical Sliced-Wasserstein (SSW) has recently been proposed to measure the discrepancy between spherical data distributions in various fields, such as geology, medical domains, computer vision, and deep representation learning. However, in the original SSW, all projection directions are treated equally, which is too idealistic and cannot accurately reflect the importance of different projection directions for various data distributions. To address this issue, we propose a novel data-adaptive Discriminative Spherical Sliced-Wasserstein (DSSW) distance, which utilizes a projected energy function to determine the discriminative projection direction for SSW. In our new DSSW, we introduce two types of projected energy functions to generate the weights for projection directions with complete theoretical guarantees. The first type employs a non-parametric deterministic function that transforms the projected Wasserstein distance into its corresponding weight in each projection direction. This improves the performance of the original SSW distance with negligible additional computational overhead. The second type utilizes a neural network-induced function that learns the projection direction weight through a parameterized neural network based on data projections. This further enhances the performance of the original SSW distance with less extra computational overhead. Finally, we evaluate the performance of our proposed DSSW by comparing it with several state-of-the-art methods across a variety of machine learning tasks, including gradient flows, density estimation on real earth data, and self-supervised learning.