Synchronized Optimal Transport for Joint Modeling of Dynamics Across Multiple Spaces

Optimal transport has been an essential tool for reconstructing dynamics from complex data. With the increasingly available multifaceted data, a system can often be characterized across multiple spaces. Therefore, it is crucial to maintain coherence in the dynamics across these diverse spaces. To address this challenge, we introduce Synchronized Optimal Transport (SyncOT), a novel approach to jointly model dynamics that represent the same system through multiple spaces. With given correspondence between the spaces, SyncOT minimizes the aggregated cost of the dynamics induced across all considered spaces. The problem is discretized into a finite-dimensional convex problem using a staggered grid. Primal-dual algorithm-based approaches are then developed to solve the discretized problem. Various numerical experiments demonstrate the capabilities and properties of SyncOT and validate the effectiveness of the proposed algorithms.