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

2025-06-10

AI Summary

This paper addresses uncertainty quantification in regression for functional and high-dimensional complex responses (e.g., functions, random graphs) in separable Hilbert spaces. We propose a model-free, conditional depth-based method for constructing predictive regions. Our approach innovatively integrates kernel mean embedding with integral probability metrics to establish the first computationally tractable conditional depth framework. We further introduce a conformalized variant that yields predictive/tolerance regions with finite-sample, non-asymptotic marginal coverage guarantees. Theoretically, we establish both conditional and unconditional consistency as well as optimal convergence rates. In simulations across diverse functional and Euclidean data settings, our method significantly improves coverage accuracy and region compactness. Applied to digital health motion behavior analysis, it enables reliable, individualized uncertainty characterization for personalized recommendations.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

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

Problem

• Develops model-free uncertainty quantification for regression in Hilbert spaces
• Introduces conformal prediction for finite-sample guarantees in depth measures
• Enhances prediction regions using kernel mean embeddings for faster convergence

Innovation

• Model-free kernel conformal depth measures
• Conditional kernel mean embeddings
• Conformal prediction for finite samples