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A general formulation of reweighted least squares fitting

Carlotta Giannelli, Sofia Imperatore, Lisa Maria Kreusser, Estefanía Loayza-Romero, Fatemeh Mohammadi, Nelly Villamizar

TL;DR

This work generalizes weighted least squares fitting by expressing any finite-dimensional WLS solution as a convex combination of interpolants, with weights given by $\\lambda_K = \omega_K |B_K|^2$. It then develops a robust, IRLS-inspired strategy that updates data weights to either emphasize sharp features or suppress outliers, and applies it to spline spaces including hierarchical and THB-splines for adaptive curve and surface fitting. The authors demonstrate, through polynomial and spline experiments, that reweighting improves feature preservation and outlier handling, often achieving better accuracy with fewer degrees of freedom than traditional LS or smoothing splines. Overall, the method provides a principled framework for adaptive, marker-informed fitting in high-dimensional domains, with potential extensions to time-series data and automatic marker classification.

Abstract

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.

A general formulation of reweighted least squares fitting

TL;DR

This work generalizes weighted least squares fitting by expressing any finite-dimensional WLS solution as a convex combination of interpolants, with weights given by . It then develops a robust, IRLS-inspired strategy that updates data weights to either emphasize sharp features or suppress outliers, and applies it to spline spaces including hierarchical and THB-splines for adaptive curve and surface fitting. The authors demonstrate, through polynomial and spline experiments, that reweighting improves feature preservation and outlier handling, often achieving better accuracy with fewer degrees of freedom than traditional LS or smoothing splines. Overall, the method provides a principled framework for adaptive, marker-informed fitting in high-dimensional domains, with potential extensions to time-series data and automatic marker classification.

Abstract

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when the solution is sought in any finite-dimensional vector space; secondly, to provide a general strategy to iteratively update the weights according to the approximation error and apply it to the spline fitting problem. In the experiments, we provide numerical examples for the case of polynomials and splines spaces. Subsequently, we evaluate the performance of our fitting scheme for spline curve and surface approximation, including adaptive spline constructions.
Paper Structure (17 sections, 2 theorems, 23 equations, 7 figures, 2 tables)

This paper contains 17 sections, 2 theorems, 23 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Interpolation and weighted least squares
  3. Weighted least squares via interpolation
  4. Consequences of the interpolatory formulation of weighted least squares approximation
  5. Hierarchical spline models
  6. Verification of \ref{['th:genweightedsum']} for spline models
  7. Weighted least squares with hierarchical box splines
  8. Reweighted least squares spline fitting
  9. Reweighted least squares adaptive spline fitting
  10. Numerical experiments
  11. Reweighted spline curve fitting with type I markers and comparison with ordinary least squares
  12. Reweighted spline curve fitting with type I markers and comparison with smoothing splines
  13. Reweighted adaptive spline fitting for type I markers
  14. Reweighted adaptive spline fitting for type II markers
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 1

The weighted least squares approximant $v\in V$ of the set of points $\bigl\{(\bm x_i,f_i)\bigr\}_{i=1}^m$ is the weighted sum of the interpolants $v_K\in V$ for $K\in\mathcal{P}_{n}^{\star}$ which interpolate the points $(\bm x_i,f_i)$ for $i\in K$, i.e.,

Figures (7)

  • Figure 1: Numerical verification of \ref{['eq:001']}, \ref{['th:genweightedsum']}: $v_K(\bm x)$ interpolants for polynomials (left) and splines (center) and the final weighted least squares approximations as sum of interpolants (right) for polynomials (red) and splines (blue).
  • Figure 2: Hierarchical box spline mesh (left); weighted least squares surfaces for $\omega_\gamma=6$ (center) and 100 (right).
  • Figure 3: Point cloud (blue dots) and set of type I markers (black dots)
  • Figure 4: Curve fitting experiment described in \ref{['subsec:num_exp_rWLS-landmarks']} using ordinary LS (red) and rWLS (purple) for data with markers of type I.
  • Figure 5: Curve fitting experiment described in \ref{['subsec:num_exp_rWLS-smoothing']} using smoothing spline (red) and rWLS (purple) for data with markers of type I (black dots).
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3