Modified least squares method and a review of its applications in machine learning and fractional differential/integral equations

TL;DR

The paper addresses improving least squares curve fitting by introducing a modified method based on fractional polynomials in Müntz spaces $M_n^{\lambda}$. It constructs orthogonal fractional bases $L_i(x;\lambda,W)$ through a weight-aware three-term recurrence, enabling direct coefficient computation and mitigating ill-conditioning. The approach is validated on synthetic fractional-power functions, fractional differential/integral equation examples, and regression tasks, showing improved accuracy, robustness to noise, and applicability to ML. The work highlights the potential of fractional polynomial features and orthogonal bases to enhance ML pipelines and fractional equation solvers.

Abstract

The least squares method provides the best-fit curve by minimizing the total squares error. In this work, we provide the modified least squares method based on the fractional orthogonal polynomials that belong to the space $M_{n}^λ := \text{span}\{1,x^λ,x^{2λ},\ldots,x^{nλ}\},~λ\in (0,2]$. Numerical experiments demonstrate how to solve different problems using the modified least squares method. Moreover, the results show the advantage of the modified least squares method compared to the classical least squares method. Furthermore, we discuss the various applications of the modified least squares method in the fields like fractional differential/integral equations and machine learning.

Figures (1)

• Figure 1: Exact solution and approximate solution for Example \ref{['6ex8']} with $\lambda = 0.75$ and different values of $n$.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Remark 1
• Remark 2
• Theorem 4.1
• Theorem 4.2
• Example 1
• Example 2
• Example 3
• Example 4