A Refined Algorithm for Curve Fitting by Segmented Straight Lines

TL;DR

This paper introduces an efficient, exact dynamic programming algorithm for piecewise-linear curve fitting that improves upon previous methods with superpolynomial complexity, enabling practical applications.

Contribution

It presents a novel hybrid dynamic programming algorithm for exact curve fitting with a small number of segments, reducing computational complexity.

Findings

The algorithm achieves polynomial time complexity.

It provides exact solutions for least squares approximation.

It outperforms previous mixed integer programming methods.

Abstract

We consider least squares approximation of a function of one variable by a continuous, piecewise-linear approximand that has a small number of breakpoints. This problem was notably considered by Bellman who proposed an approximate algorithm based on dynamic programming. Many suboptimal approaches have been suggested, but so far, the only exact methods resort to mixed integer programming with superpolynomial complexity growth. In this paper, we present an exact and efficient algorithm based on dynamic programming with a hybrid value function. The achieved time-complexity seems to be polynomial.