标题： 度量空间中的共形和kNN预测不确定性量化算法 显示英文标题

标题： Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces

摘要： 本文介绍了一种在度量空间中定义的回归模型的不确定性量化框架。 利用一种新定义的同方差性概念，我们开发了一种保真预测算法，该算法提供有限样本覆盖保证和oracle估计量的快速收敛率。 在异方差情况下，我们放弃这些非渐近保证以获得统计效率，提出了一种无需保真校准的局部$k$-最近邻方法，该方法适应于每个特定非线性空间的几何结构。 两种过程都可以与任何回归算法一起使用，并且可以扩展到大型数据集，使从业者能够插入他们首选的模型并融入领域知识。 我们在最小条件下证明了所提出的估计量的一致性。 最后，我们在涉及随机响应对象（如概率分布和图拉普拉斯）的个性化医学应用中展示了我们方法的实用性。 显示英文摘要

摘要： This paper introduces a framework for uncertainty quantification in regression models defined in metric spaces. Leveraging a newly defined notion of homoscedasticity, we develop a conformal prediction algorithm that offers finite-sample coverage guarantees and fast convergence rates of the oracle estimator. In heteroscedastic settings, we forgo these non-asymptotic guarantees to gain statistical efficiency, proposing a local $k$--nearest--neighbor method without conformal calibration that is adaptive to the geometry of each particular nonlinear space. Both procedures work with any regression algorithm and are scalable to large data sets, allowing practitioners to plug in their preferred models and incorporate domain expertise. We prove consistency for the proposed estimators under minimal conditions. Finally, we demonstrate the practical utility of our approach in personalized--medicine applications involving random response objects such as probability distributions and graph Laplacians.