Synchronized Optimal Transport for Joint Modeling of Dynamics Across Multiple Spaces

TL;DR

This paper introduces Synchronized Optimal Transport (SyncOT), a new method for jointly modeling system dynamics across multiple related spaces, ensuring coherence and leveraging correspondence between these spaces.

Contribution

The paper proposes SyncOT, a novel approach that synchronizes dynamics modeling across multiple spaces using optimal transport, with a discretized convex formulation and primal-dual algorithms.

Findings

SyncOT effectively models dynamics across multiple spaces.

The proposed algorithms solve the discretized problem efficiently.

Numerical experiments validate the method's effectiveness.

Abstract

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.