Greedy algorithms for learning via exponential-polynomial splines

TL;DR

This paper introduces adaptive greedy algorithms for learning with exponential-polynomial splines, aiming to reduce overfitting and oscillations by selecting optimal sampling points based on residuals or approximation error bounds.

Contribution

It extends kernel-based greedy algorithms to exponential-polynomial splines and proposes two adaptive methods for selecting spline nodes to improve approximation quality.

Findings

The $f$-greedy algorithm adapts to specific target functions.

The $\lambda$-greedy algorithm selects nodes independently of target functions.

Both methods enhance spline approximation by adaptive node selection.

Abstract

Kernel-based schemes are state-of-the-art techniques for learning by data. In this work we extend some ideas about kernel-based greedy algorithms to exponential-polynomial splines, whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals ($f$-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ($\lambda$-greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e. the spline nodes. However, while the {$f$-greedy} selection is tailored to one specific target function, the $\lambda$-greedy algorithm is independent of the function values and enables us to define a priori optimal interpolation nodes.