# Multivariate Standardized Residuals for Conformal Prediction

arXiv: https://arxiv.org/abs/2507.20941 · HTML: https://mlsanity.com/paper/2507.20941/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 49 references that are in the map.
- First posted on arXiv 28 Jul 2025.

## Builds on (most cited first, 24 of 24)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [What's a good imputation to predict with missing values?](https://mlsanity.com/paper/2106.00311/) (2021)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Conformal Uncertainty Sets for Robust Optimization](https://mlsanity.com/paper/2105.14957/) (2021)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Conformal Prediction with Missing Values](https://mlsanity.com/paper/2306.02732/) (2023)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Probabilistic Conformal Prediction with Approximate Conditional Validity](https://mlsanity.com/paper/2407.01794/) (2024)
- [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025)
- [Conformal Temporal Logic Planning using Large Language Models](https://mlsanity.com/paper/2309.10092/) (2023)
- [Optimal Transport-based Conformal Prediction](https://mlsanity.com/paper/2501.18991/) (2025)
- [Multivariate Quantile Function Forecaster](https://mlsanity.com/paper/2202.11316/) (2022)
- [Rectifying Conformity Scores for Better Conditional Coverage](https://mlsanity.com/paper/2502.16336/) (2025)
- [Flexible Conformal Highest Predictive Conditional Density Sets](https://mlsanity.com/paper/2406.18052/) (2024)
- [Conformal Prediction with Partially Labeled Data](https://mlsanity.com/paper/2306.01191/) (2023)
- [Conformal Prediction for Hierarchical Data](https://mlsanity.com/paper/2411.13479/) (2024)
- [Predictive Uncertainty Quantification with Missing Covariates](https://mlsanity.com/paper/2405.15641/) (2024)
- [JAPAN: Joint Adaptive Prediction Areas with Normalising-Flows](https://mlsanity.com/paper/2505.23196/) (2025)
- [Conformal Prediction with Corrupted Labels: Uncertain Imputation and Robust Re-weighting](https://mlsanity.com/paper/2505.04733/) (2025)
- [Fair Conformal Prediction for Incomplete Covariate Data](https://mlsanity.com/paper/2504.12582/) (2025)

## Built on by (most cited first, 6 of 6)

- [Conditional Coverage Diagnostics for Conformal Prediction](https://mlsanity.com/paper/2512.11779/) (2025)
- [Colorful Pinball: Density-Weighted Quantile Regression for Conditional Guarantee of Conformal Prediction](https://mlsanity.com/paper/2512.24139/) (2025)
- [Interpretable Multivariate Conformal Prediction with Fast Transductive Standardization](https://mlsanity.com/paper/2512.15383/) (2025)
- [Super-Level-Set Regression: Conditional Quantiles via Volume Minimization](https://mlsanity.com/paper/2605.06210/) (2026)
- [Tail-Aware Geometry Learning for Conformal Ellipsoids](https://mlsanity.com/paper/2609.27221/) (2026)
- [Multivariate conformal uncertainty propagation in multitask atomistic simulation: Successes and pitfalls](https://mlsanity.com/paper/2609.31384/) (2026)

## Abstract (arXiv)

While split conformal prediction guarantees marginal coverage, approaching the stronger property of conditional coverage is essential for reliable uncertainty quantification. Naive conformal scores, however, suffer from poor conditional coverage in heteroskedastic settings. In univariate regression, this is commonly addressed by normalizing non-conformity scores using an estimated local score variance. In this work, we propose a natural extension of this normalization to the multivariate setting, effectively whitening the residuals to decouple output correlations and standardize local variance. Furthermore, we derive a sufficient condition characterizing a broad class of distributions for which standardized residuals yield asymptotic conditional coverage. We demonstrate that using the Mahalanobis distance induced by a learned local covariance as a non-conformity score provides a closed-form, computationally efficient mechanism for capturing inter-output correlations and heteroskedasticity, avoiding the expensive sampling required by previous methods based on cumulative distribution functions. This structure unlocks several practical extensions, including the handling of missing output values, the refinement of conformal sets when partial information is revealed, and the construction of valid conformal sets for transformations of the output. Finally, we provide extensive empirical evidence on both synthetic and real-world datasets showing that our approach yields conformal sets that improve upon the conditional coverage of existing multivariate baselines.