Time Parameterized Optimal Transport

Abstract: Optimal transport has gained significant attention in recent years due to its effectiveness in deep learning and computer vision. Its descendant metric, the Wasserstein distance, has been particularly successful in measuring distribution dissimilarities. While extensive research has focused on optimal transport and its regularized variants (such as entropy, sparsity, and capacity constraints) the role of time has been largely overlooked. However, time is a critical factor in real world transport problems. In this work, we introduce a time parameterized formulation of the optimal transport problem, incorporating a time variable t to represent sequential steps and enforcing specific constraints at each step. We propose a systematic method to solve a special subproblem and develop a heuristic search algorithm that achieves nearly optimal solutions while significantly reducing computational time.