Multivariate conformal prediction requires nonconformity scores that compress residual vectors into scalars while preserving certain implicit geometric structure of the residual distribution. We introduce a Multivariate Kernel Score (MKS) that produces prediction regions that explicitly adapt to this geometry. We show that the proposed score resembles the Gaussian process posterior variance, unifying Bayesian uncertainty quantification with the coverage guarantees of frequentist-type. Moreover, the MKS can be decomposed into an anisotropic Maximum Mean Discrepancy (MMD) that interpolates between kernel density estimation and covariance-weighted distance. We prove finite-sample coverage guarantees and establish convergence rates that depend on the effective rank of the kernel-based covariance operator rather than the ambient dimension, enabling dimension-free adaptation. On regression tasks, the MKS reduces the volume of prediction region significantly, compared to ellipsoidal baselines while maintaining nominal coverage, with larger gains at higher dimensions and tighter coverage levels.

翻译：多元共形预测需要非一致性评分，将残差向量压缩为标量，同时保留残差分布的隐式几何结构。我们引入一种多元核评分（MKS），其产生的预测区域能够显式地适配这种几何结构。我们证明，所提评分类似于高斯过程后验方差，从而将贝叶斯不确定性量化与频率学派类型的覆盖保证统一起来。此外，MKS可分解为各向异性的最大均值差异（MMD），该差异在核密度估计与协方差加权距离之间进行插值。我们证明了有限样本覆盖保证，并建立了依赖于核基础上协方差算子有效秩（而非环境维度）的收敛速率，从而实现了维度无关的自适应。在回归任务上，与椭球基线方法相比，MKS在维持名义覆盖的前提下显著减小了预测区域的体积，且在更高维度和更紧的覆盖水平下获得更大收益。