Energy-Based Sliced Wasserstein Distance
This work addresses a computational and statistical bottleneck in optimal transport metrics for machine learning, offering a parameter-free and stable alternative to existing methods, though it appears incremental as it builds directly on sliced Wasserstein distance frameworks.
The paper tackled the problem of choosing an effective slicing distribution for the sliced Wasserstein distance, proposing an energy-based distribution to overcome limitations of prior fixed or optimized parametric distributions, resulting in a novel metric (EBSW) that showed favorable performance in experiments on tasks like point-cloud gradient flow and color transfer.
The sliced Wasserstein (SW) distance has been widely recognized as a statistically effective and computationally efficient metric between two probability measures. A key component of the SW distance is the slicing distribution. There are two existing approaches for choosing this distribution. The first approach is using a fixed prior distribution. The second approach is optimizing for the best distribution which belongs to a parametric family of distributions and can maximize the expected distance. However, both approaches have their limitations. A fixed prior distribution is non-informative in terms of highlighting projecting directions that can discriminate two general probability measures. Doing optimization for the best distribution is often expensive and unstable. Moreover, designing the parametric family of the candidate distribution could be easily misspecified. To address the issues, we propose to design the slicing distribution as an energy-based distribution that is parameter-free and has the density proportional to an energy function of the projected one-dimensional Wasserstein distance. We then derive a novel sliced Wasserstein metric, energy-based sliced Waserstein (EBSW) distance, and investigate its topological, statistical, and computational properties via importance sampling, sampling importance resampling, and Markov Chain methods. Finally, we conduct experiments on point-cloud gradient flow, color transfer, and point-cloud reconstruction to show the favorable performance of the EBSW.