用熵正则化最优传输构造条件分布映射，实现无参数条件采样。
Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps
- 基于熵正则化最优传输，构建无参数条件Brenier映射估计器。
- 在高斯设定下精确刻画缩放参数与样本数的关系，提升估计稳定性。
- 相比传统方法，在基准数据集和贝叶斯推断中表现更优，适合复杂分布建模。
条件采样是统计建模中的基础任务：从联合分布的有限数据点出发，生成条件分布的样本。一种有前景的方法是构建条件Brenier映射，其将参考分布通过映射转化为目标分布的条件分布。尽管已有多种估计器，但很少有具备统计或算法保证。为此，我们提出一种基于熵正则化最优传输的无参数条件Brenier映射估计器。该方法利用Carlier等（2010）的结果：在缩放二次代价下，最优传输映射渐近收敛于条件Brenier映射；我们的估计器正是这类收敛映射的熵正则化版本。我们在高斯设定下完全刻画了代价缩放参数与样本数之间的关系，给出选择该参数的启发式依据。最后，我们通过对比基准数据集和贝叶斯推断问题中其他机器学习与非参数方法，验证了该估计器的性能优势。
原文摘要 · Abstract (English)
Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of \emph{entropic} optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems.
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