Optimal Flow Transport and its Entropic Regularization: a GPU-friendly Matrix Iterative Algorithm for Flow Balance Satisfaction
Liangliang Shi Yufeng Li Kaipeng Zeng Yihui Tu Junchi Yan
The Sinkhorn algorithm, based on Entropic Regularized Optimal Transport (OT), has garnered significant attention due to its computational efficiency enabled by GPU-friendly matrix-vector multiplications. However, vanilla OT primarily deals with computations between the source and target nodes in a bipartite graph, limiting its practical application in real-world transportation scenarios.
In this paper, we introduce the concept of Optimal Flow Transport (OFT) as an extension, where we consider a more general graph setting and the marginal constraints in vanilla OT are replaced by flow balance constraints. 
To obtain solutions, we incorporate entropic regularization into the OFT and introduce virtual flows for individual nodes to tackle the issue of potentially numerous isolated nodes lacking flow passages. Our proposition, the OFT-Sinkhorn algorithm, utilizes GPU-friendly matrix iterations to maintain flow balance constraints and minimize the objective function, and theoretical results for global convergence is also proposed in this paper.
Furthermore, we enhance OFT by introducing capacity constraints on nodes and edges, transforming the OFT problem into a minimum-cost flow problem. We then present the Capacity-Constrained EOFT-Sinkhorn algorithm and compare it with the traditional Minimum cost flow (MCF) algorithm, showing that our algorithm is quite efficient for calculation. 
In particular, our EOFT-Sinkhorn is evaluated on high-precision and integer-precision MCF problems with different scales from one hundred to five thousand size, exhibiting significant time efficiency and the ability to approximate optimal solutions.
Optimal Transport, Flow Balance Constraints, Sinkhorn Algorithm, Network Flow Theory