MLOT: Extending the Bipartite Structure towards Multi-Layered Structure for Optimal Transport

Liangliang Shi, Rui Tao, Junchi Yan

Keywords: Optimal Transport, Multi-Layered Structure, Entropic Regularization, Sinkhorn Algorithm

TL;DR: We extend the bipartite structure of OT into multi-layed OT, validate its great efficiency. We further reformulate Text-Image retrieval task into MLOT problem, which obtain increase in recall.

Abstract: Despite its remarkable success and widespread adoption in various domains, optimal transport (OT) has a rather simple structure, relying on bipartite graphs with only two layers of nodes for transportation. In this paper, we propose a multi-layered OT approach that extends the original two-layer structure to handle transportation problems across multiple hierarchical levels. Within this framework, the source distribution flows through intermediate layers, before reaching the target distribution. Unlike previous variants of OT that involve multiple distributions, our multi-layered OT typically involves uncertain intermediate distributions, which need to be computed based on the relationships between the preceding and succeeding distributions. Under entropic regularization, MLOT-Sinkhorn algorithm is further proposed for multi-layered OT, which can be accelerated using GPUs and significantly outperforms general solvers such as Gurobi. The theoretical results of our entropic MLOT are also given in this paper. In the experiments, we validate its speed advantage and convergence performance. We further validate its feasibility through Text-Image retrieval and intermediate image computing task, which demonstrates reformulating the problems as MLOT can achieve better results. Source code will be made available.

Primary Area: optimization

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Submission Number: 3699