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 a generalization 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, it fits naturally in the unbalanced optimal transport problem setting as well.

翻译：在最优传输领域，两个主流分支相互对立：（i）无正则化最优传输（“a-la-Kantorovich”型），其传输方案极度稀疏但算法扩展性差；（ii）熵正则化最优传输（“a-la-Sinkhorn-Cuturi”型），其拥有近线性近似算法但传输方案极度非稀疏。本文证明，将后者推广至温度指数测度——一种具有间接测度归一化的指数族推广——可实现极佳的折中方案：既具有极快的近似算法，又能控制稀疏性直至稀疏模式。此外，该框架亦能自然适配非平衡最优传输问题场景。