In this presentation, we propose a unified data-driven framework that can learn interaction cost function from noisy and incomplete empirical matching data through optimal transport theory. We use entropy to regularize the Wasserstein distance used in the inverse optimal transport problem, develop an algorithm to compute its solution. The learned cost can predict new matching. The talk is based on joint work with Ruilin Li (GT Math), Xiaojing Ye (GSU, Math) and Hongyuan Zha (GT, CSE).