A Taylor Series Approach to Correction of Input Errors in Gaussian Process Regression

2025-04-25

AI Summary

In Gaussian process (GP) regression, systematic input bias errors—e.g., from drifting mobile sensor localization—degrade prediction accuracy; existing approaches require full model retraining upon input correction, incurring prohibitive computational cost. Method: We propose a training-free online GP model refinement method that dynamically corrects for time-varying input biases. Leveraging the differentiability of the squared-exponential kernel, we introduce a second-order Taylor expansion of the GP predictive mean and variance with respect to input perturbations, efficiently computed via pre-estimated Jacobian and Hessian matrices of the kernel. Input bias is estimated in real time using a Kalman filter. Contribution/Results: The method significantly improves both point prediction accuracy and uncertainty quantification quality. In two simulation studies, it achieves millisecond-scale model refinement without retraining, while relaxing the restrictive assumption of zero-mean input noise inherent in conventional GP formulations.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Gaussian Processes (GPs) are widely recognized as powerful non-parametric models for regression and classification. Traditional GP frameworks predominantly operate under the assumption that the inputs are either accurately known or subject to zero-mean noise. However, several real-world applications such as mobile sensors have imperfect localization, leading to inputs with biased errors. These biases can typically be estimated through measurements collected over time using, for example, Kalman filters. To avoid recomputation of the entire GP model when better estimates of the inputs used in the training data become available, we introduce a technique for updating a trained GP model to incorporate updated estimates of the inputs. By leveraging the differentiability of the mean and covariance functions derived from the squared exponential kernel, a second-order correction algorithm is developed to update the trained GP models. Precomputed Jacobians and Hessians of kernels enable real-time refinement of the mean and covariance predictions. The efficacy of the developed approach is demonstrated using two simulation studies, with error analyses revealing improvements in both predictive accuracy and uncertainty quantification.

Problem

• Corrects biased input errors in Gaussian Process Regression
• Updates trained GP models without full recomputation
• Improves predictive accuracy and uncertainty quantification

Innovation

• Taylor series corrects input errors in GP
• Second-order algorithm updates trained GP models
• Precomputed Jacobians enable real-time refinement