Piecewise Convex-Concave Approximation in the $\ell_{\infty}$ Norm

TL;DR

This paper introduces a novel piecewise convex-concave approximation method in the infinity norm, providing an efficient algorithm for smooth data approximation with controlled curvature changes, suitable for large and noisy datasets.

Contribution

It develops a new optimization-based approach with an efficient $O(nq)$ algorithm for piecewise convex-concave approximation under infinity norm constraints.

Findings

Effective for large, densely-packed data

Produces good approximations even with large errors

Operates efficiently with linear complexity

Abstract

Suppose that $\ff \in \reals^{n}$ is a vector of $n$ error-contaminated measurements of $n$ smooth values measured at distinct and strictly ascending abscissae. The following projective technique is proposed for obtaining a vector of smooth approximations to these values. Find \yy\ minimizing $\| \yy - \ff \|_{\infty}$ subject to the constraints that the second order consecutive divided differences of the components of \yy\ change sign at most $q$ times. This optimization problem (which is also of general geometrical interest) does not suffer from the disadvantage of the existence of purely local minima and allows a solution to be constructed in $O(nq)$ operations. A new algorithm for doing this is developed and its effectiveness is proved. Some of the results of applying it to undulating and peaky data are presented, showing that it is economical and can give very good results, particularly for large densely-packed data, even when the errors are quite large.