用点到最近邻距离定义鲁棒性得分，提升预测集在异常值下的可靠性。
Conformal Robust Set Estimation
- 基于点到第⌊n/2⌋+1近邻的距离设计非一致性评分，增强鲁棒性。
- 无论样本量大小，都能保证边际有效性，且与理想中心集收敛。
- 适用于重尾或多重模态分布，适合对稳定性要求高的场景。
置信预测在可交换性假设下提供有限样本、无需分布假设的覆盖性，但标准方法在存在异常值或重尾时可能缺乏鲁棒性。本文提出一种基于非一致性评分的鲁棒置信预测方法，该评分定义为某点周围的半质量半径，等价于其第⌊n/2⌋+1近邻的距离。我们证明所得置信区域对任意样本量均具有边际有效性，并以概率收敛于通过距离到测度函数定义的鲁棒总体中心集。在温和正则条件下，建立了指数集中性和尾部界，量化了经验置信区域与其总体对应物之间的偏差。这些结果为在重尾或多重模态分布中使用鲁棒几何评分提供了概率依据。
原文摘要 · Abstract (English)
Conformal prediction provides finite-sample, distribution-free coverage under exchangeability, but standard constructions may lack robustness in the presence of outliers or heavy tails. We propose a robust conformal method based on a non-conformity score defined as the half-mass radius around a point, equivalently the distance to its $(\lfloor n/2\rfloor+1)$-nearest neighbour. We show that the resulting conformal regions are marginally valid for any sample size and converge in probability to a robust population central set defined through a distance-to-a-measure functional. Under mild regularity conditions, we establish exponential concentration and tail bounds that quantify the deviation between the empirical conformal region and its population counterpart. These results provide a probabilistic justification for using robust geometric scores in conformal prediction, even for heavy-tailed or multi-modal distributions.
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