# CD-split and HPD-split: efficient conformal regions in high dimensions

arXiv: https://arxiv.org/abs/2007.12778 · HTML: https://mlsanity.com/paper/2007.12778/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2020.

## Built on by (most cited first, 30 of 76)

- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Improving Conditional Coverage via Orthogonal Quantile Regression](https://mlsanity.com/paper/2106.00394/) (2021)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [Predictive Inference with Feature Conformal Prediction](https://mlsanity.com/paper/2210.00173/) (2022)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Efficient and Differentiable Conformal Prediction with General Function Classes](https://mlsanity.com/paper/2202.11091/) (2022)
- [Conformal Prediction with Missing Values](https://mlsanity.com/paper/2306.02732/) (2023)
- [Conformal Contextual Robust Optimization](https://mlsanity.com/paper/2310.10003/) (2023)
- [Conformal Prediction with Learned Features](https://mlsanity.com/paper/2404.17487/) (2024)
- [Split Localized Conformal Prediction](https://mlsanity.com/paper/2206.13092/) (2022)
- [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)
- [Optimal Transport-based Conformal Prediction](https://mlsanity.com/paper/2501.18991/) (2025)
- [Conformal Prediction via Regression-as-Classification](https://mlsanity.com/paper/2404.08168/) (2024)
- [Few-Shot Calibration of Set Predictors via Meta-Learned Cross-Validation-Based Conformal Prediction](https://mlsanity.com/paper/2210.03067/) (2022)
- [Rectifying Conformity Scores for Better Conditional Coverage](https://mlsanity.com/paper/2502.16336/) (2025)
- [Regression Trees for Fast and Adaptive Prediction Intervals](https://mlsanity.com/paper/2402.07357/) (2024)
- [Conformal Thresholded Intervals for Efficient Regression](https://mlsanity.com/paper/2407.14495/) (2024)
- [Towards Instance-Wise Calibration: Local Amortized Diagnostics and Reshaping of Conditional Densities (LADaR)](https://mlsanity.com/paper/2205.14568/) (2022)
- [Volume Optimality in Conformal Prediction with Structured Prediction Sets](https://mlsanity.com/paper/2502.16658/) (2025)
- [Epistemic Uncertainty in Conformal Scores: A Unified Approach](https://mlsanity.com/paper/2502.06995/) (2025)
- [Conditional Coverage Diagnostics for Conformal Prediction](https://mlsanity.com/paper/2512.11779/) (2025)
- [Distribution-Free Conformal Joint Prediction Regions for Neural Marked Temporal Point Processes](https://mlsanity.com/paper/2401.04612/) (2024)
- [Conditional Conformal Risk Adaptation](https://mlsanity.com/paper/2504.07611/) (2025)
- [Volume-Sorted Prediction Set: Efficient Conformal Prediction for Multi-Target Regression](https://mlsanity.com/paper/2503.02205/) (2025)
- [Multivariate Standardized Residuals for Conformal Prediction](https://mlsanity.com/paper/2507.20941/) (2025)
- [CP4SBI: Local Conformal Calibration of Credible Sets in Simulation-Based Inference](https://mlsanity.com/paper/2508.17077/) (2025)
- [MD-split+: Practical Local Conformal Inference in High Dimensions](https://mlsanity.com/paper/2107.03280/) (2021)
- [Conformalized Deep Splines for Optimal and Efficient Prediction Sets](https://mlsanity.com/paper/2311.00774/) (2023)
- [Toward Conditional Distribution Calibration in Survival Prediction](https://mlsanity.com/paper/2410.20579/) (2024)

## Abstract (arXiv)

Conformal methods create prediction bands that control average coverage assuming solely i.i.d. data. Although the literature has mostly focused on prediction intervals, more general regions can often better represent uncertainty. For instance, a bimodal target is better represented by the union of two intervals. Such prediction regions are obtained by CD-split , which combines the split method and a data-driven partition of the feature space which scales to high dimensions. CD-split however contains many tuning parameters, and their role is not clear. In this paper, we provide new insights on CD-split by exploring its theoretical properties. In particular, we show that CD-split converges asymptotically to the oracle highest predictive density set and satisfies local and asymptotic conditional validity. We also present simulations that show how to tune CD-split. Finally, we introduce HPD-split, a variation of CD-split that requires less tuning, and show that it shares the same theoretical guarantees as CD-split. In a wide variety of our simulations, CD-split and HPD-split have better conditional coverage and yield smaller prediction regions than other methods.