# Exact and Approximate Conformal Inference for Multi-Output Regression

arXiv: https://arxiv.org/abs/2210.17405 · HTML: https://mlsanity.com/paper/2210.17405/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 40 references that are in the map.
- First posted on arXiv 31 Oct 2022.

## Builds on (most cited first, 16 of 16)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [Learning by Transduction](https://mlsanity.com/paper/1301.7375/) (2013)
- [Conformal Prediction: a Unified Review of Theory and New Challenges](https://mlsanity.com/paper/2005.07972/) (2020)
- [A Conformal Prediction Approach to Explore Functional Data](https://mlsanity.com/paper/1302.6452/) (2013)
- [Copula-based conformal prediction for Multi-Target Regression](https://mlsanity.com/paper/2101.12002/) (2021)
- [Conformal Uncertainty Sets for Robust Optimization](https://mlsanity.com/paper/2105.14957/) (2021)
- [Exchangeability, Conformal Prediction, and Rank Tests](https://mlsanity.com/paper/2005.06095/) (2020)
- [Conformal Prediction Bands for Multivariate Functional Data](https://mlsanity.com/paper/2106.01792/) (2021)
- [Fast Exact Conformalization of Lasso using Piecewise Linear Homotopy](https://mlsanity.com/paper/1708.00427/) (2017)
- [Computing Full Conformal Prediction Set with Approximate Homotopy](https://mlsanity.com/paper/1909.09365/) (2019)
- [Stable Conformal Prediction Sets](https://mlsanity.com/paper/2112.10224/) (2021)
- [Concepts and Applications of Conformal Prediction in Computational Drug Discovery](https://mlsanity.com/paper/1908.03569/) (2019)
- [Exact Optimization of Conformal Predictors via Incremental and Decremental Learning](https://mlsanity.com/paper/2102.03236/) (2021)
- [Root-finding Approaches for Computing Conformal Prediction Set](https://mlsanity.com/paper/2104.06648/) (2021)
- [Constructing Prediction Intervals with Neural Networks: An Empirical Evaluation of Bootstrapping and Conformal Inference Methods](https://mlsanity.com/paper/2210.05354/) (2022)

## Built on by (most cited first, 7 of 7)

- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Interpretable Multivariate Conformal Prediction with Fast Transductive Standardization](https://mlsanity.com/paper/2512.15383/) (2025)
- [Conformal Reliability: A New Evaluation Metric for Conditional Generation](https://mlsanity.com/paper/2605.30807/) (2026)
- [SPACE: Sample-cloud Predictive Adaptive Conformal Ellipsoids for Multivariate Time-Series Forecasting](https://mlsanity.com/paper/2608.17333/) (2026)
- [Common-Center Geometry and Certified Radial Reconstruction for Energy-Form Full Conformal Regions](https://mlsanity.com/paper/2608.24964/) (2026)
- [Exact Risk-Complexity Laws for Projective Boundaries in Scenario Optimization and Distribution-Free Certification](https://mlsanity.com/paper/2609.01355/) (2026)
- [SimplexUQ: An Evaluation Framework and Benchmark for Conformal Uncertainty on Simplex-Valued Predictions](https://mlsanity.com/paper/2610.00523/) (2026)

## Abstract (arXiv)

It is common in machine learning to estimate a response y given covariate information x. However, these predictions alone do not quantify any uncertainty associated with said predictions. One way to overcome this deficiency is with conformal inference methods, which construct a set containing the unobserved response y with a prescribed probability. Unfortunately, even with a one-dimensional response, conformal inference is computationally expensive despite recent encouraging advances. In this paper, we explore multi-output regression, delivering exact derivations of conformal inference p-values when the predictive model can be described as a linear function of y. Additionally, we propose textttunionCP and a multivariate extension of textttrootCP as efficient ways of approximating the conformal prediction region for a wide array of multi-output predictors, both linear and nonlinear, while preserving computational advantages. We also provide both theoretical and empirical evidence of the effectiveness of these methods using both real-world and simulated data.