We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal algorithm with a closed-form projection and proximal operator scheme, thereby affording a more scalable algorithm for computing optimal transport plans. We provide a novel argument for the uniqueness of the optimum even in the absence of strong convexity. Our experiments show that the new regularizer not only results in a better preservation of the class structure in the data but also yields additional robustness to the data geometry, compared to previous regularizers.

翻译：我们发展了一个新颖的理论框架，用于理解尊重类结构的最优传输方案。为此，我们提出了一种带有范数和正则化项的凸最优传输规划，该规划在几何假设下可证明地恢复潜在的类结构。此外，我们推导出一个具有闭式投影和近端算子方案的加速近端算法，从而提供了一种计算最优传输计划的更具可扩展性的算法。即使在缺乏强凸性的情况下，我们也为最优解的唯一性提供了新颖的论证。实验表明，与之前的正则化方法相比，新的正则化器不仅更好地保留了数据中的类结构，而且对数据几何结构具有额外的鲁棒性。