Optimal transport (OT) formalizes the problem of finding an optimal coupling
between probability measures given a cost matrix. The inverse problem of
inferring the cost given a coupling is Inverse Optimal Transport (IOT). IOT is
less well understood than OT. We formalize and systematically analyze the
properties of IOT using tools from the study of entropy-regularized OT.
Theoretical contributions include characterization of the manifold of
cross-ratio equivalent costs, the implications of model priors, and derivation
of an MCMC sampler. Empirical contributions include visualizations of
cross-ratio equivalent effect on basic examples and simulations validating
theoretical results.

本文通过熵正则化的 Optimal transport（OT）工具，对 Inverse Optimal Transport（IOT）进行正式化和系统分析，包括代价等价成本的流形特性、模型先验的影响以及基于模拟的结果验证等方面。

离散概率逆最优输运

Discrete Probabilistic Inverse Optimal Transport

Inverse optimal transport (OT) refers to the problem of learning the cost
function for OT from observed transport plan or its samples. In this paper, we
derive an unconstrained convex optimization formulation of the inverse OT
problem, which can be further augmented by any customizable regularization. We
provide a comprehensive characterization of the properties of inverse OT,
including uniqueness of solutions. We also develop two numerical algorithms,
one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for
discrete OT, and the other one is a learning based algorithm that parameterizes
the cost function as a deep neural network for continuous OT. The novel
framework proposed in the work avoids repeatedly solving a forward OT in each
iteration which has been a thorny computational bottleneck for the bi-level
optimization in existing inverse OT approaches. Numerical results demonstrate
promising efficiency and accuracy advantages of the proposed algorithms over
existing state-of-the-art methods.

本文提出了一种无约束凸优化形式的逆向最优输运问题，其中包括了两种数字算法，并使用深度神经网络参数化成本函数，以此解决了现有逆向最优输运方法中前向计算瓶颈的问题，并展示了这种方法的高效性和准确性。