# Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces

arXiv: https://arxiv.org/abs/2506.08325 · HTML: https://mlsanity.com/paper/2506.08325/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 88 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls](https://mlsanity.com/paper/1712.09089/) (2017)
- [Choosing among notions of multivariate depth statistics](https://mlsanity.com/paper/2004.01927/) (2020)
- [Distributional data analysis of accelerometer data from the NHANES database using nonparametric survey regression models](https://mlsanity.com/paper/2104.01165/) (2021)
- [An Efficient Doubly-Robust Test for the Kernel Treatment Effect](https://mlsanity.com/paper/2304.13237/) (2023)
- [Depth for curve data and applications](https://mlsanity.com/paper/1901.00180/) (2019)
- [Multilevel functional distributional models with application to continuous glucose monitoring in diabetes clinical trials](https://mlsanity.com/paper/2403.10514/) (2024)
- [Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis](https://mlsanity.com/paper/2105.12778/) (2021)
- [Extremal Depth for Functional Data and Applications](https://mlsanity.com/paper/1511.00128/) (2015)
- [On Mahalanobis distance in functional settings](https://mlsanity.com/paper/1803.06550/) (2018)
- [Nonparametric imputation by data depth](https://mlsanity.com/paper/1701.03513/) (2017)
- [Statistical depth in abstract metric spaces](https://mlsanity.com/paper/2107.13779/) (2021)
- [A uniform kernel trick for high-dimensional two-sample problems](https://mlsanity.com/paper/2210.02171/) (2022)
- [Conditional Mean and Variance Estimation via textitk-NN Algorithm with Automated Variance Selection](https://mlsanity.com/paper/2402.01635/) (2024)

## Abstract (arXiv)

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations