在沃尔瑟斯坦空间上实现熵正则泛函的线性收敛优化
Linear convergence of proximal descent schemes on the Wasserstein space
- 基于最小化移动框架设计近端下降算法
- 在平坦凸性假设下证明线性收敛，无需测地凸性
- 适合研究最优传输与扩散过程的理论工作者
我们研究受Jordan、Kinderlehrer和Otto提出的最小化移动方案启发的近端下降方法，用于优化沃尔瑟斯坦空间上的熵正则泛函。在平坦凸性假设下建立了线性收敛性，从而放宽了对测地凸性的依赖。分析避免了离散时间演化变分不等式（EVI）的适应需求，转而利用统一的对数索博列夫不等式（LSI）和熵“夹逼”引理，扩展了arXiv:2201.10469和arXiv:2202.01009的分析。证明中的主要挑战在于：通过证明迭代序列属于特定Sobolev正则类，确保相对费希尔信息在每一步均有定义。由于相对熵在沃尔瑟斯坦空间上不可微，我们进一步证明其具有唯一的沃尔瑟斯坦次梯度，且费希尔信息有限。
原文摘要 · Abstract (English)
We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space. We establish linear convergence under flat convexity assumptions, thereby relaxing the common reliance on geodesic convexity. Our analysis circumvents the need for discrete-time adaptations of the Evolution Variational Inequality (EVI). Instead, we leverage a uniform logarithmic Sobolev inequality (LSI) and the entropy ``sandwich" lemma, extending the analysis from arXiv:2201.10469 and arXiv:2202.01009. The major challenge in the proof via LSI is to show that the relative Fisher information is well-defined at every step of the scheme. Since the relative entropy is not Wasserstein differentiable, we prove that along the scheme the iterates belong to a certain class of Sobolev regularity, and hence the relative entropy has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite.
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