Extraction of implicit field cost via inverse optimal Schrödinger bridge

Path planning and navigation are typically conducted based on a field cost defined over a given space. However, determining an appropriate field cost is challenging. Animals, including humans, act based on past experience. Thus, their behaviour can be considered to include implicit field costs based on their experiences or accumulated knowledge. We address the extraction of such field costs using an approach based on optimal transport. The optimal transport problem aims to derive the most efficient transport plan based on a given field cost, initial distribution, and terminal distribution. Thus, in this study, we investigate the extraction of such a field cost via an inverse optimal transport problem. Although the optimal transport problem has been studied extensively with respect to its objective function and transport plan, research on the inverse problem remains scarce. In particular, there has been notably limited research on the inverse problem in dynamic optimal transport. Because of the complexity of the inverse problem, we propose a method that leverages the gradients commonly employed in deep learning, which are well-suited to optimal transport. Moreover, we propose an efficient solution that leverages the Schrödinger bridge, which is equivalent to the optimal transport problem under certain conditions. In addition, using behavioural data from 176 pigeons, we demonstrate through simulations that our method successfully extracts the field costs embedded in their movements.