String Diagram of Optimal Transports

TL;DR

This work introduces a hierarchical framework for optimal transport based on string diagrams, enabling complex hierarchical transports to be reduced to standard OT problems via sequential and parallel cost-matrix compositions. It develops duality theory for composed OTs, defines aligned string diagrams, and proves a reduction to monolithic OT that preserves the problem’s cost structure. An algorithm is proposed to compute optimal hierarchical transportation plans by solving a monolithic OT and then synthesizing the hierarchical solution, with extensions to OTs that include choices in cost matrices. Empirical results demonstrate that the proposed monolithic approach outperforms naive LP methods, highlighting practical efficiency and scalability for structured OT problems in hierarchical planning contexts.

Abstract

We present a novel hierarchical framework for optimal transport (OT) using string diagrams, namely string diagrams of optimal transports. This framework reduces complex hierarchical OT problems to standard OT problems, allowing efficient synthesis of optimal hierarchical transportation plans. Our approach uses algebraic compositions of cost matrices to effectively model hierarchical structures. We also study an adversarial situation with multiple choices in the cost matrices, where we present a polynomial-time algorithm for a relaxation of the problem. Experimental results confirm the efficiency and performance advantages of our proposed algorithm over the naive method.

Figures (4)

• Figure 1: A string diagram $\mathbb{D}\coloneqq \mathcal{A}\fatsemi (\mathcal{B}\otimes\mathcal{C})$ with distributions $\mathbf{a}, \mathbf{b}$.
• Figure 2: An example of our reduction given by \ref{['cor:correctRD']}. The string diagram $\mathbb{D}$ in the above figure is formally given by $\mathbb{D}\coloneqq \mathcal{A}\fatsemi (\mathcal{B}\otimes \mathrm{id}_{k^{\mathcal{A}}})\fatsemi (\mathcal{C}\otimes \mathcal{D})$ with the sequential composition $\fatsemi$, the parallel composition $\otimes$, and the identity $\mathrm{id}_{k^{\mathcal{A}}}$. Similarly, the cost matrix $\mathbf{C}^{\mathbb{D}}$ on the right is given by $\mathbf{C}^{\mathbb{D}}\coloneqq \mathbf{C}^{\mathcal{A}}\fatsemi (\mathbf{C}^{\mathcal{B}}\otimes \mathbf{id}_{k^{\mathcal{A}}})\fatsemi (\mathbf{C}^{\mathcal{C}}\otimes \mathbf{C}^{\mathcal{D}})$ with the compositions $\fatsemi,\otimes$, and the idenity cost matrix $\mathbf{id}_{k^{\mathcal{A}}}$.
• Figure 3: The typing rules for string diagrams.
• Figure 4: Influence of the complexity of the algebraic structures. See Discussion for explanations.

Theorems & Definitions (26)

• Definition 2.1: OT
• Definition 2.2: sequentially composed OT
• Definition 2.3: parallelly composed OT
• Definition 3.1
• Proposition 3.2: strong duality
• Proposition 3.3
• Definition 3.4
• Proposition 3.5: strong duality
• Proposition 3.6
• Definition 4.1