通过最优控制估计无配对数据转换中的薛定谔势，实现高效密度迁移。
Tight Bounds for Schrödinger Potential Estimation in Unpaired Data Translation
- 以奥恩斯坦-乌伦贝克过程为参考，估计薛定谔势以实现最优密度转换。
- 在高斯混合势类上，理论给出紧的泛化误差界，收敛速度接近快速率。
- 适合无配对数据生成与翻译任务，尤其适用于分布转移建模场景。
基于薛定谔桥与随机最优控制理论的现代生成建模与无配对数据转换方法旨在以最优方式将初始分布转换为目标分布。本文假设仅能获取初始与目标分布的独立同分布样本，适用于生成建模与无配对数据翻译。基于随机最优控制框架，选择奥恩斯坦-乌伦贝克过程作为参考过程，并估计对应的薛定谔势。引入以耦合间KL散度为风险函数，推导出在薛定谔势类（包括高斯混合）上经验风险最小化器的紧泛化误差界。得益于奥恩斯坦-乌伦贝克过程的混合性质，在有利情况下几乎达到快速收敛速率，仅受对数因子影响。数值实验验证了所提方法的有效性。
原文摘要 · Abstract (English)
Modern methods of generative modelling and unpaired data translation based on Schrödinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from the initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we choose an Ornstein-Uhlenbeck process as the reference one and estimate the corresponding Schrödinger potential. Introducing a risk function as the Kullback-Leibler divergence between couplings, we derive tight bounds on the generalization ability of an empirical risk minimizer over a class of Schrödinger potentials, including Gaussian mixtures. Thanks to the mixing properties of the Ornstein-Uhlenbeck process, we almost achieve fast rates of convergence, up to some logarithmic factors, in favourable scenarios. We also illustrate the performance of the suggested approach with numerical experiments.
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