Core Concepts
Solutions to the entropically regularized optimal transport problem can be characterized by a well-posed ordinary differential equation (ODE). This ODE-based approach provides a new numerical method to solve the optimal transport problem and allows for the computation of derivatives of the optimal cost in the fully regularized limit.
Abstract
The content presents a characterization of the solution to the entropically regularized optimal transport problem using a well-posed ordinary differential equation (ODE). The approach works for discrete marginals and general cost functions, and applies to multi-marginal problems and those with additional linear constraints.
The key highlights and insights are:
The authors derive an ODE that governs the behavior of the optimal dual Kantorovich potential as a function of the regularization parameter. This ODE can be used to efficiently compute the entire curve of solutions, interpolating between the product measure and the optimal transport solution.
The ODE formulation allows for the computation of derivatives of the optimal cost in the fully regularized limit (η → ∞), which has received relatively limited attention compared to expansions around the unregularized limit.
The authors provide a general framework for applying the Sinkhorn algorithm to linearly constrained optimal transport problems, unifying and extending previous work on particular cases.
Several numerical examples are presented, demonstrating the feasibility of the ODE-based method and comparing it to the traditional Sinkhorn algorithm. The examples include two-marginal optimal transport, multi-marginal optimal transport, and martingale optimal transport.
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