In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, "\`a-la-Kantorovich", which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, "\`a-la-Sinkhorn-Cuturi", which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.

翻译：在最优传输领域中，两个重要分支相互对立：(i) 无正则化的Kantorovich型最优传输，其传输方案极其稀疏但算法扩展性较差；(ii) 熵正则化的Sinkhorn-Cuturi型最优传输，虽具有近线性近似算法但会导致传输方案极度非稀疏。本文证明，将后者扩展至温度指数测度（一种具有间接测度归一化的指数族推广形式）可达到理想的折中状态：既能实现极快近似算法，又能控制稀疏性直至特定的稀疏模式。此外，我们的公式自然适配非平衡最优传输问题的设定。