Anchor-Based Function Extrapolation with Proven Bounds and Projection Guarantees

TL;DR

This paper introduces a framework using anchor functions to improve and certify the accuracy of function extrapolation beyond sampled domains, providing rigorous bounds and probabilistic guarantees.

Contribution

It develops a model-agnostic approach that recasts extrapolation as a feasibility problem, offering certified bounds and a projection method to enhance baseline approximations.

Findings

Significant reduction in extrapolation error in experiments.

Established new stability constants for extrapolation.

Validated theoretical bounds with numerical experiments.

Abstract

Classical approximation and learning methods are typically optimized for interpolation over a sampled domain {\Omega}, with no guarantees on their behavior in an extrapolation region {\Xi}, where small in-domain errors may amplify. We develop a model-agnostic framework that recasts extrapolation as a feasibility and projection problem with rigorous guarantees. The approach is built around anchor functions, auxiliary constructions for which one can certify an upper bound on the {\Xi}-distance to the unknown target function. Such certificates define feasible sets that are proven to contain the true function. Given any baseline approximation (e.g., least-squares or regularized regression), we obtain a corrected extrapolation by projecting the baseline onto the feasible set; the resulting predictor is proven not to increase the error on {\Xi}, and we prove quantitative bounds on the improvement. We establish new stability constants governing extrapolation, including a tight spectral condition number and a numerically stable inner-domain bound that connects in-domain error to extrapolation risk. To reduce conservatism of worst-case certification, we also propose probabilistic anchor functions that yield high-confidence feasible sets. Numerical experiments, including geomagnetic field modeling and nonlinear oscillators, demonstrate substantial reductions in extrapolation error and corroborate the theoretical predictions.