论文标题
具有一般凸正则化的最佳运输损失和凹痕算法
Optimal Transport losses and Sinkhorn algorithm with general convex regularization
论文作者
论文摘要
我们引入了一类新的凸定规范化的最佳运输损失，该损失概括了最佳运输和sindhorn差异的经典熵验证，并提出了一种广义的sndhorn算法。我们的框架统一了许多先前出现在文献中的正规化和数值方法。我们显示了双重问题，互补的懈度条件的最大化器的存在，为这些变异问题提供了完整的解决方案的特征。结果，我们研究了这些损失的结构性特性，包括连续性，可不同性，并为其梯度提供明确的公式。最后，即使在连续的环境中，我们也提供了广义沉没算法的融合和稳定性的理论保证。此处开发的技术也直接适用于研究Wasserstein Barycenters或更一般的多核心问题。
We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework unifies many regularizations and numerical methods previously appeared in the literature. We show the existence of the maximizer for the dual problem, complementary slackness conditions, providing a complete characterization of solutions for such class of variational problems. As a consequence, we study structural properties of these losses, including continuity, differentiability and provide explicit formulas for the its gradient. Finally, we provide theoretical guarantees of convergences and stability of the generalized Sinkhorn algorithm, even in the continuous setting. The techniques developed here are directly applicable also to study Wasserstein barycenters or, more generally, multi-marginal problems.
