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A Kernel Nonconformity Score for Multivariate Conformal Prediction

NeurIPS 2026 · Learning theory and algorithms › Prediction and decision losses · approach: Conformal prediction sets · 0 citations (OpenAlex)

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Proposes the Multivariate Kernel Score (MKS), a kernel generalization of the regularized Mahalanobis nonconformity score for multivariate conformal prediction, shown to equal a GP posterior variance and decompose into an MMD term and a KPCA correction.

Problem

Nonconformity scores for multivariate conformal prediction compress residual vectors into scalars; ellipsoidal scores over-inflate regions for non-elliptical residuals, while density-based scores ignore directional structure and degrade in high dimension.

Approach

Define the nonconformity score as a regularized Mahalanobis distance in an RKHS using the centered kernel covariance operator; this equals a GP posterior variance on calibration locations, decomposes into MMD (density) plus a KPCA anisotropy correction, and reduces to the Mahalanobis distance for a linear kernel.

Key results

• On synthetic data (linear model, alpha=0.1), MKS volume 2.48 vs Mahalanobis 2.82 and Bonferroni 4.14, with nominal coverage 0.899.
• On real data, MKS volume ratio to Mahalanobis ranges from about 0.25 to 0.95 across datasets, dimensions and coverage levels (e.g. House d=4 at 0.90: 0.61), while coverage tracks nominal levels.
• Volumes are 9 to 31% smaller than Mahalanobis on synthetic data, with gains growing at tighter coverage levels.

Surprising findings

• The density-only MMD score produces larger regions than the Mahalanobis ellipsoid under both models on synthetic data.
• The MKS score is a function of calibration locations only, not calibration responses, despite being interpreted as a GP posterior variance.

Limitations

• Experiments primarily use RBF kernel with heuristic hyperparameters; kernel and lengthscale choice deferred to future work.
• Coverage theorem for conditional guarantee relies on i.i.d. and exogeneity assumptions; marginal coverage under alpha-mixing only in appendix.
• Assumptions such as RKHS estimation rate are not verified for RBF kernel and the WSC metric favors linear-slab-aligned regions.

Experimental setup

Datasets: synthetic bivariate regression (n=50,000) House Bio Blog

Benchmarks: synthetic bivariate regression House Bio Blog

Metrics: coverage volume worst-slab coverage volume ratio to Mahalanobis

Baselines: Mahalanobis score Bonferroni method density-only MMD score HPD-split

Builds on: split conformal prediction Mahalanobis nonconformity score maximum mean discrepancy kernel PCA Gaussian processes

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