We propose a new general framework for recovering low-rank structure in optimal transport using Schatten-$p$ norm regularization. Our approach extends existing methods that promote sparse and interpretable transport maps or plans, while providing a unified and principled family of convex programs that encourage low-dimensional structure. The convexity of our formulation enables direct theoretical analysis: we derive optimality conditions and prove recovery guarantees for low-rank couplings and barycentric maps in simplified settings. To efficiently solve the proposed program, we develop a mirror descent algorithm with convergence guarantees for $p \geq 1$. Experiments on synthetic and real data demonstrate the method's efficiency, scalability, and ability to recover low-rank transport structures.

翻译：我们提出了一种利用Schatten-$p$范数正则化来恢复最优传输中低秩结构的新通用框架。我们的方法扩展了现有那些旨在促进稀疏且可解释的传输映射或方案的技术，同时提供了一个统一且具有原则性的凸优化程序族，以鼓励低维结构。我们公式的凸性使得直接的理论分析成为可能：我们推导了最优性条件，并在简化设定下证明了低秩耦合与重心映射的恢复保证。为了高效求解所提出的优化程序，我们开发了一种镜像下降算法，该算法对 $p \geq 1$ 的情况具有收敛保证。在合成数据与真实数据上的实验证明了该方法的高效性、可扩展性以及恢复低秩传输结构的能力。