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Ordinary Differential Equation for Entropic Optimal Transport and Its Linearly Constrained Variants


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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Deeper Inquiries

How can the ODE-based approach be extended to handle general (non-discrete) marginals

To extend the ODE-based approach to handle general (non-discrete) marginals, we can consider representing the marginals as continuous probability distributions. This would involve formulating the ODE in terms of probability density functions instead of discrete measures. The key challenge would be in discretizing the continuous distributions to make them amenable to numerical methods. One approach could be to use kernel density estimation or other smoothing techniques to approximate the continuous marginals with discrete points. Additionally, the ODE formulation would need to be adapted to handle the continuous nature of the marginals, possibly involving integral equations and functional analysis techniques.

What are the potential applications and implications of the ability to compute higher-order derivatives of the optimal cost in the fully regularized limit

The ability to compute higher-order derivatives of the optimal cost in the fully regularized limit can have several important applications and implications. Firstly, higher-order derivatives can provide more detailed information about the behavior of the optimal cost function near the fully regularized limit. This can be valuable for understanding the curvature and shape of the cost function, which can in turn inform optimization strategies and convergence properties of numerical algorithms. Furthermore, higher-order derivatives can be used to derive more accurate Taylor expansions of the cost function, enabling better approximations and predictions of the cost behavior for different regularization levels. This can be particularly useful in sensitivity analysis and optimization under uncertainty, where a precise understanding of the cost function's behavior is crucial.

How might the insights and techniques developed in this work be applied to other optimization problems beyond optimal transport

The insights and techniques developed in this work on entropic optimal transport and ODE-based numerical methods have broad applications beyond the specific problem domain. One potential application is in machine learning and data science, where optimal transport is used in tasks such as domain adaptation, image registration, and generative modeling. By extending the ODE approach to handle more general optimization problems, these techniques could be applied to a wide range of optimization tasks in various fields, including economics, logistics, and computer science. The ability to efficiently compute solutions and derivatives of cost functions can enhance the performance of optimization algorithms and enable more accurate modeling and analysis of complex systems. Additionally, the ODE-based approach could be integrated into existing optimization frameworks to improve the scalability and efficiency of solving large-scale optimization problems.
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