A Fast Proximal Point Method for Computing Exact Wasserstein Distance
This addresses a bottleneck in machine learning and related fields where approximate methods degrade performance, offering a more stable and exact solution.
The authors tackled the high computational complexity and numerical instability of computing exact Wasserstein distances by developing an Inexact Proximal point method (IPOT), which converges to exact distances with theoretical guarantees, similar complexity to Sinkhorn, and avoids performance degradation in applications like generative models.
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developing an Inexact Proximal point method for exact Optimal Transport problem (IPOT) with the proximal operator approximately evaluated at each iteration using projections to the probability simplex. The algorithm (a) converges to exact Wasserstein distance with theoretical guarantee and robust regularization parameter selection, (b) alleviates numerical stability issue, (c) has similar computational complexity to Sinkhorn, and (d) avoids the shrinking problem when apply to generative models. Furthermore, a new algorithm is proposed based on IPOT to obtain sharper Wasserstein barycenter.