Simplifying Optimal Transport through Schatten-$p$ Regularization
This work addresses the challenge of simplifying optimal transport for applications requiring low-dimensional structure, though it appears incremental as it extends existing methods with a unified regularization approach.
The authors tackled the problem of recovering low-rank structure in optimal transport by proposing a framework using Schatten-$p$ norm regularization, resulting in a convex formulation with theoretical guarantees and an efficient mirror descent algorithm that demonstrated scalability and recovery of low-rank structures in experiments.
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.