A Smoothed Dual Approach for Variational Wasserstein Problems
This work addresses a computational bottleneck for researchers and practitioners in machine learning and optimization dealing with Wasserstein-based variational problems, offering an incremental improvement over existing methods.
The paper tackles the computational challenge of variational problems involving Wasserstein distances by introducing a smoothed dual approach, resulting in simpler implementation and improved numerical stability for applications like Wasserstein barycenters and gradient flows.
Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals.