用施瓦茨-p范数正则化，高效恢复最优传输的低秩结构。
Simplifying Optimal Transport through Schatten-$p$ Regularization
- 引入施瓦茨-p范数正则化，统一建模低秩传输结构。
- 理论证明可恢复低秩耦合与重心映射，具收敛性保障。
- 适用于需要简洁、可解释传输路径的科研与工程场景。
我们提出一种新框架，通过施瓦茨-p范数正则化恢复最优传输中的低秩结构。该方法拓展了以往促进稀疏性和可解释性的传输图或计划的方法，提供了一类统一且有原则的凸优化问题，以鼓励低维结构。由于公式具有凸性，可直接进行理论分析：在简化设定下，我们推导出最优性条件，并证明了对低秩耦合与重心映射的恢复保证。为高效求解该问题，我们开发了一种镜面下降算法，对p ≥ 1具备收敛性保证。在合成数据与真实数据上的实验表明，该方法具有高效性、可扩展性，且能有效恢复低秩传输结构。
原文摘要 · Abstract (English)
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.
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