标题： 生成式AI的薛定谔桥：软约束公式和收敛性分析 显示英文标题

标题： Schrödinger bridge for generative AI: Soft-constrained formulation and convergence analysis

摘要： 生成式人工智能可以被表述为学习一个将简单参考测度映射到复杂数据分布的模型的问题，并且由于它们在通过熵正则化随机动力学在预设边缘之间进行插值方面的共同性质，最近与经典的Schrödinger桥问题（SBPs）有了强烈的联系。 然而，经典的SBP施加了硬终端约束，这在实际实现中经常导致不稳定性，尤其是在高维或数据稀缺的场景中。 为了解决这个挑战，我们遵循所谓的软约束Schrödinger桥问题（SCSBP）的想法，在这种情况下，终端约束被一个一般的惩罚函数所取代。 这种放松导致了一个更灵活的麦肯-弗拉索夫类型的随机控制公式。 我们证明了所有惩罚水平下最优解的存在性，并证明了随着惩罚的增长，控制和价值函数以线性速率收敛到经典SBP的控制和价值函数。 我们的分析基于Doob的h变换表示、Schrödinger势的稳定性结果、Gamma收敛，以及一个新颖的固定点论证，该论证将测度空间上的优化问题与一个辅助熵最优传输问题耦合在一起。 这些结果不仅为软约束桥提供了第一个定量收敛保证，还揭示了惩罚正则化如何促进稳健的生成建模、微调和迁移学习。 显示英文摘要

摘要： Generative AI can be framed as the problem of learning a model that maps simple reference measures into complex data distributions, and it has recently found a strong connection to the classical theory of the Schr\"odinger bridge problems (SBPs) due partly to their common nature of interpolating between prescribed marginals via entropy-regularized stochastic dynamics. However, the classical SBP enforces hard terminal constraints, which often leads to instability in practical implementations, especially in high-dimensional or data-scarce regimes. To address this challenge, we follow the idea of the so-called soft-constrained Schr\"odinger bridge problem (SCSBP), in which the terminal constraint is replaced by a general penalty function. This relaxation leads to a more flexible stochastic control formulation of McKean-Vlasov type. We establish the existence of optimal solutions for all penalty levels and prove that, as the penalty grows, both the controls and value functions converge to those of the classical SBP at a linear rate. Our analysis builds on Doob's h-transform representations, the stability results of Schr\"odinger potentials, Gamma-convergence, and a novel fixed-point argument that couples an optimization problem over the space of measures with an auxiliary entropic optimal transport problem. These results not only provide the first quantitative convergence guarantees for soft-constrained bridges but also shed light on how penalty regularization enables robust generative modeling, fine-tuning, and transfer learning.