# Multi-Modal Conformal Prediction Regions with Simple Structures by Optimizing Convex Shape Templates

arXiv: https://arxiv.org/abs/2312.07434 · HTML: https://mlsanity.com/paper/2312.07434/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 38 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 21 of 21)

- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Safe Planning in Dynamic Environments using Conformal Prediction](https://mlsanity.com/paper/2210.10254/) (2022)
- [Learning Optimal Conformal Classifiers](https://mlsanity.com/paper/2110.09192/) (2021)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Conformal Prediction Under Feedback Covariate Shift for Biomolecular Design](https://mlsanity.com/paper/2202.03613/) (2022)
- [Adaptive Conformal Prediction for Motion Planning among Dynamic Agents](https://mlsanity.com/paper/2212.00278/) (2022)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Sample-Efficient Safety Assurances using Conformal Prediction](https://mlsanity.com/paper/2109.14082/) (2021)
- [Conformal Prediction for STL Runtime Verification](https://mlsanity.com/paper/2211.01539/) (2022)
- [Conformal Prediction Regions for Time Series using Linear Complementarity Programming](https://mlsanity.com/paper/2304.01075/) (2023)
- [iDECODe: In-distribution Equivariance for Conformal Out-of-distribution Detection](https://mlsanity.com/paper/2201.02331/) (2022)
- [Efficient and Differentiable Conformal Prediction with General Function Classes](https://mlsanity.com/paper/2202.11091/) (2022)
- [Split Localized Conformal Prediction](https://mlsanity.com/paper/2206.13092/) (2022)
- [Forking Uncertainties: Reliable Prediction and Model Predictive Control with Sequence Models via Conformal Risk Control](https://mlsanity.com/paper/2310.10299/) (2023)
- [Physics Constrained Motion Prediction with Uncertainty Quantification](https://mlsanity.com/paper/2302.01060/) (2023)
- [Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inference](https://mlsanity.com/paper/2309.09187/) (2023)
- [Conformalized Multimodal Uncertainty Regression and Reasoning](https://mlsanity.com/paper/2309.11018/) (2023)
- [Reliable Prediction Intervals with Directly Optimized Inductive Conformal Regression for Deep Learning](https://mlsanity.com/paper/2302.00872/) (2023)

## Built on by (most cited first, 4 of 4)

- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Learnable Conformal Prediction with Context-Aware Nonconformity Functions for Robotic Planning and Perception](https://mlsanity.com/paper/2509.21955/) (2025)
- [Conformal Prediction for Generative Models via Adaptive Cluster-Based Density Estimation](https://mlsanity.com/paper/2601.22298/) (2026)
- [Conformal Safety Monitoring for Flight Testing: A Case Study in Data-Driven Safety Learning](https://mlsanity.com/paper/2511.20811/) (2025)

## Abstract (arXiv)

Conformal prediction is a statistical tool for producing prediction regions for machine learning models that are valid with high probability. A key component of conformal prediction algorithms is a non-conformity score function that quantifies how different a model's prediction is from the unknown ground truth value. Essentially, these functions determine the shape and the size of the conformal prediction regions. While prior work has gone into creating score functions that produce multi-model prediction regions, such regions are generally too complex for use in downstream planning and control problems. We propose a method that optimizes parameterized shape template functions over calibration data, which results in non-conformity score functions that produce prediction regions with minimum volume. Our approach results in prediction regions that are multi-modal, so they can properly capture residuals of distributions that have multiple modes, and practical, so each region is convex and can be easily incorporated into downstream tasks, such as a motion planner using conformal prediction regions. Our method applies to general supervised learning tasks, while we illustrate its use in time-series prediction. We provide a toolbox and present illustrative case studies of F16 fighter jets and autonomous vehicles, showing an up to 68% reduction in prediction region area compared to a circular baseline region.