Approximate Discrete Optimal Transport Plan with Auxiliary Measure Method

Optimal transport (OT) between two measures plays an essential role in many fields, ranging from economy, biology to machine learning and artificial intelligence. Conventional discrete OT problem can be solved using linear programming (LP). Unfortunately, due to the large scale and the intrinsic non-linearity, achieving discrete OT plan with adequate accuracy and efficiency is challenging. Generally speaking, the OT plan is highly sparse. This work proposes an auxiliary measure method to use the semi-discrete OT maps to estimate the sparsity of the discrete OT plan with squared Euclidean cost. Although obtaining the accurate semi-discrete OT maps is difficult, we can find the sparsity information through computing the approximate semi-discrete OT maps by convex optimization. The sparsity information can be further incorporated into the downstream LP optimization to greatly reduce the computational complexity and improve the accuracy. We also give a theoretic error bound between the estimated transport plan and the OT plan in terms of Wasserstein distance. Experiments on both synthetic data and color transfer tasks demonstrate the accuracy and efficiency of the proposed method.