While the continuous Entropic Optimal Transport (EOT) field has been actively developing in recent years, it became evident that the classic EOT problem is prone to different issues like the sensitivity to outliers and imbalance of classes in the source and target measures. This fact inspired the development of solvers that deal with the unbalanced EOT (UEOT) problem $-$ the generalization of EOT allowing for mitigating the mentioned issues by relaxing the marginal constraints. Surprisingly, it turns out that the existing solvers are either based on heuristic principles or heavy-weighted with complex optimization objectives involving several neural networks. We address this challenge and propose a novel theoretically-justified, lightweight, unbalanced EOT solver. Our advancement consists of developing a novel view on the optimization of the UEOT problem yielding tractable and a non-minimax optimization objective. We show that combined with a light parametrization recently proposed in the field our objective leads to a fast, simple, and effective solver which allows solving the continuous UEOT problem in minutes on CPU. We prove that our solver provides a universal approximation of UEOT solutions and obtain its generalization bounds. We give illustrative examples of the solver's performance.

翻译：尽管连续熵最优传输领域近年来发展活跃，但经典EOT问题易受多种问题影响的特性已日益明显，例如对异常值的敏感性以及源测度与目标测度间的类别不平衡。这一事实推动了非平衡EOT求解器的发展——通过放宽边缘约束来缓解上述问题的EOT泛化形式。令人惊讶的是，现有求解器要么基于启发式原理，要么因涉及多个神经网络的复杂优化目标而显得笨重。我们针对这一挑战提出了一种新颖的、理论依据充分的轻量级非平衡EOT求解器。我们的进展在于提出了对UEOT问题优化的新视角，从而得到可处理的非极小极大优化目标。结合该领域近期提出的轻量参数化方法，我们证明该目标可形成快速、简单且有效的求解器，能在CPU上数分钟内解决连续UEOT问题。我们证明了该求解器能实现UEOT解的通用逼近，并获得了其泛化边界。我们给出了该求解器性能的示例说明。