Anchor-Based Function Extrapolation with Proven Bounds and Projection Guarantees

TL;DR

A model-agnostic framework that recasts extrapolation as a feasibility and projection problem with rigorous guarantees is developed, and new stability constants governing extrapolation are established, including a tight spectral condition number and a numerically stable inner-domain bound that connects in-domain error to extrapolation risk.

Abstract

Classical approximation and learning methods are typically optimized for interpolation over a sampled domain Ω, with no guarantees on their behavior in an extrapolation region Ξ, 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 Ξ-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 Ξ, 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.

Figures (10)

• Figure 1: Geometric configuration used in Lemma \ref{['lemma:circle_point_circle_with_bounds']} illustrating the projection of an exterior point onto a feasible ball and the resulting distance reduction.
• Figure 2: True oscillator (black), LASSO fit (solid orange), projected fit (dashed orange), and anchor bounds (blue, $\pm 1.2$ around 0). The vertical line marks the split between $\Omega$ and $\Xi$. Projection toward the constant anchor substantially reduces extrapolation error.
• Figure 3: Extrapolation behavior over $\Xi$ for LS, LASSO ($\alpha=0.001$), and the projected solution of Theorem \ref{['theorem:5']}. Left: distance–orientation plot relative to the LS approximation; the dashed circle denotes the feasible-set boundary. Right: true function (black), LS (red), LASSO (orange), and projection (green). Reported root errors on $\Xi$ are LS: 4.39, LASSO: 3.51, Projection: 2.00, with improvement bounds $0.30 \le \Delta E_\Xi \le 1.54$.
• Figure 4: LS and ridge regression fits on $\Omega$ and their projected counterparts on $\Xi$ for $B_r(\mu)$ (degree $8$, $\sigma=0.05$, $\kappa_{\mathrm{spec}}=309$). Projection substantially reduces extrapolation error on $\Xi$ for this real-world geomagnetic dataset.
• Figure 5: Comparison between upper bounds calculated using Theorem \ref{['theorem:all_kappa_calculation']} ($\kappa$) and Theorem \ref{['theorem:all_kappa_calculation_improved']} ($\kappa_{\mathrm{spec}}$). The plots represent $15$ basis functions (right), $10$ basis functions (middle), and $5$ basis functions (left) with a log-scaled y-axis.

Theorems & Definitions (42)

• Remark 2.1: More general function spaces
• Definition 2.2: Extrapolation problem
• Definition 2.3: Domain error and root error
• Definition 2.4: Anchor functions
• Definition 2.5: Anchored extrapolation problem
• Remark 2.6: Certified anchor tolerance
• Definition 3.1: anchor feasible set
• Definition 3.2: minimal feasible set
• Lemma 3.3
• proof