This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions discretized over a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive $c$-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

翻译：本文提出了一种用于Wasserstein重心的无约束凹对偶新形式。通过将标准对偶梯度上升算法适配至Sobolev几何结构，我们推导出一种可扩展的Sobolev梯度上升（SGA）算法，用于计算在规则网格上离散化的输入分布的重心。尽管算法简洁，我们提供了全局收敛性分析，其收敛速率与欧氏空间中用于最小化非光滑凸函数的经典次梯度下降法相同。SGA算法的一个核心特征是，无需强制在Kantorovich对偶势上施加计算代价高昂的$c$-凹投影算子即可保证收敛，从而在算法和理论上显著简化了现有所有用于计算精确重心的原始-对偶方法。我们的数值实验表明，SGA在现有最优传输重心求解器中具有优越的经验性能。