# Conformal inference for regression on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2310.08209 · HTML: https://mlsanity.com/paper/2310.08209/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 62 references that are in the map.
- First posted on arXiv 12 Oct 2023.

## Builds on (most cited first, 15 of 15)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Conformal Inference of Counterfactuals and Individual Treatment Effects](https://mlsanity.com/paper/2006.06138/) (2020)
- [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)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Conformal prediction for time series](https://mlsanity.com/paper/2010.09107/) (2020)
- [Conformal Bayesian Computation](https://mlsanity.com/paper/2106.06137/) (2021)
- [Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation](https://mlsanity.com/paper/1704.08095/) (2017)
- [Conformal Prediction Bands for Multivariate Functional Data](https://mlsanity.com/paper/2106.01792/) (2021)
- [Distribution-Free Prediction Bands for Multivariate Functional Time Series: an Application to the Italian Gas Market](https://mlsanity.com/paper/2107.00527/) (2021)
- [Strong Uniform Consistency with Rates for Kernel Density Estimators with General Kernels on Manifolds](https://mlsanity.com/paper/2007.06408/) (2020)
- [Parametric Regression on the Grassmannian](https://mlsanity.com/paper/1505.03832/) (2015)

## Built on by (most cited first, 1 of 1)

- [Coordinate Singularities Break Conformal Coverage for Gaze and Head Pose](https://mlsanity.com/paper/2607.02565/) (2026)

## Abstract (arXiv)

Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for non Euclidean data. Circular data is a classic example, but so is data in the space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by Y, resides in a manifold, and the covariable, denoted by X, lies in an Euclidean space. This extends the concepts delineated in citewaser14 to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of (X,Y), and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.