Linear least squares problems involving fixed knots polynomial splines and their singularity study

TL;DR

This paper investigates polynomial spline-based approximation problems in data processing, providing efficient criteria to determine when the associated linear systems are non-singular, thus enabling faster solution methods.

Contribution

It develops quick-to-verify conditions for non-singularity of system matrices in spline approximation problems, improving computational efficiency.

Findings

Sufficient conditions for non-singularity are established.

Verification of non-singularity is significantly faster than direct methods.

Algorithm efficiency is enhanced by selecting appropriate solution methods.

Abstract

In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject to optimisation, and so called prototype functions, whose choice is based on the application, rather than optimisation. The corresponding optimisation problems can be formulated as Linear Least Squares Problems (LLSPs). If the system matrix is non-singular, then the corresponding problem can be solved using the normal equations method, while for singular cases slower (but more robust) methods have to be used. In this paper we develop sufficient conditions for non-singularity. These conditions can be verified much faster than the direct singularity verification of the system matrices. Therefore, the algorithm efficiency can be improved by choosing a better suited method for solving the corresponding LLSPs.