基于流形几何的不确定性量化方法，提升回归预测的可靠性。
Geometry-Aware Uncertainty Quantification via Conformal Prediction on Manifolds
- 用测地线距离构造非符合性评分，捕捉流形上的真实误差
- 在球面和地磁场预测中，覆盖概率更稳定，最差情况表现更好
- 适合需要几何感知不确定性的科学建模任务
共形预测为回归提供有限样本覆盖率保证，但大多数标准方法针对欧几里得输出空间设计。当响应变量位于黎曼流形上时，欧氏残差和坐标基区域可能忽略定义有意义误差的几何结构。我们提出自适应测地线共形预测，一种简单框架，从测地线距离构建非符合性评分，并用交叉验证估计的局部预测难度进行归一化。在球面上，该方法生成面积与位置无关的测地线帽，而半径仍能适应异方差噪声。在球面合成实验和IGRF-14地磁场预测任务中，该自适应方法保持有效的边际覆盖率，降低条件覆盖率波动，并优于非自适应和坐标基基线方法的最差覆盖率。
原文摘要 · Abstract (English)
Conformal prediction gives finite-sample coverage guarantees for regression, but most standard constructions are designed for Euclidean output spaces. When the response lies on a Riemannian manifold, Euclidean residuals and coordinate-based regions can ignore the geometry that defines meaningful error. We propose adaptive geodesic conformal prediction, a simple framework that builds nonconformity scores from geodesic distances and normalizes them with a cross-validated estimate of local prediction difficulty. On the sphere, this produces geodesic caps whose area is independent of position, while their radii still adapt to heteroscedastic noise. In both a synthetic sphere experiment and an IGRF-14 geomagnetic field forecasting task, the adaptive method preserves valid marginal coverage, reduces variation in conditional coverage, and improves worst-case coverage relative to non-adaptive and coordinate-based baselines.
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