This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (BCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

翻译：本文分析了Bregman正则化下的逆最优传输（IOT）问题。我们在成本矩阵的若干结构性假设下，建立了适定性结果，包括解的存在性、唯一性（在等价类意义下）和稳定性。在计算方面，我们研究了成本矩阵具有一般约束的优化问题解的存在性，并给出了保证解存在的充分条件。此外，针对带有强凸惩罚项的问题，我们提出了一种非精确块坐标下降（BCD）方法。特别地，当惩罚项为二次型时，子问题的Hessian矩阵具有对角结构，这使得能够实现高效的逐元素牛顿更新。我们为该算法建立了线性收敛速率，并通过数值实验验证了其实际性能，包括稳定性界的验证、正则化效应的研究以及在婚姻匹配数据集上的应用。