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Comparison of 2D Regular Lattices for the CPWL Approximation of Functions

Mehrsa Pourya, Maïka Nogarotto, Michael Unser

TL;DR

The paper addresses the 2D CPWL approximation of functions using box-spline based search spaces on regular lattices. It derives explicit, grid-dependent error bounds in the Fourier domain and shows that the asymptotic error constant is minimized by hexagonal grids, establishing their optimality over Cartesian grids. By linking box splines to ReLU networks, it demonstrates a compact network interpretation with two hidden layers for CPWL construction and discusses how the error scales with grid parameters and the function’s smoothness. The results highlight the practical impact of grid geometry in CPWL-based approximation and neural-network representations, with hexagonal lattices offering superior asymptotic performance for CPWL approximants.

Abstract

We investigate the approximation error of functions with continuous and piecewise-linear (CPWL) representations. We focus on the CPWL search spaces generated by translates of box splines on two-dimensional regular lattices. We compute the approximation error in terms of the stepsize and angles that define the lattice. Our results show that hexagonal lattices are optimal, in the sense that they minimize the asymptotic approximation error.

Comparison of 2D Regular Lattices for the CPWL Approximation of Functions

TL;DR

The paper addresses the 2D CPWL approximation of functions using box-spline based search spaces on regular lattices. It derives explicit, grid-dependent error bounds in the Fourier domain and shows that the asymptotic error constant is minimized by hexagonal grids, establishing their optimality over Cartesian grids. By linking box splines to ReLU networks, it demonstrates a compact network interpretation with two hidden layers for CPWL construction and discusses how the error scales with grid parameters and the function’s smoothness. The results highlight the practical impact of grid geometry in CPWL-based approximation and neural-network representations, with hexagonal lattices offering superior asymptotic performance for CPWL approximants.

Abstract

We investigate the approximation error of functions with continuous and piecewise-linear (CPWL) representations. We focus on the CPWL search spaces generated by translates of box splines on two-dimensional regular lattices. We compute the approximation error in terms of the stepsize and angles that define the lattice. Our results show that hexagonal lattices are optimal, in the sense that they minimize the asymptotic approximation error.

Paper Structure

This paper contains 13 sections, 4 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Continuous and Piecewise-Linear (CPWL) Box Splines
  4. CPWL Search Space
  5. Parameterization of the Grid Matrix
  6. Formulation of the Problem
  7. Methods
  8. Computation of the Approximation Error
  9. Analysis of the Effect of the Grid
  10. Error for the Cartesian and Hexagonal Grids
  11. Error Computation for a Fourier Disk
  12. Box Splines as ReLU Networks
  13. Conclusion

Key Result

Proposition 1

Let $\varphi$ be the Cartesian box spline. Then, one has that

Figures (3)

  • Figure 1: Cartesian (left) and hexagonal ($\mu = ({\frac{\sqrt{3}}{2}})^{-0.5}$) (right) box splines.
  • Figure 2: Cartesian grid with ${\bf{\Xi}}_{\mathrm{Cart}} = [{\boldsymbol{\xi}}_1 \ {\boldsymbol{\xi}}_2] = T00T$ (left) and hexagonal grid with ${\bf{\Xi}}_{\mathrm{Hex}} = [{\boldsymbol{\xi}}_1 \ {\boldsymbol{\xi}}_2] = \mu T-0.5\mu T00.5\sqrt{3} \mu T$ and $\mu = ({\frac{\sqrt{3}}{2}})^{-0.5}$ (right). The highlighted region depicts the support of the corresponding box splines $B_{\Xi_{\mathrm{Cart}}}$ and $B_{\Xi_{\mathrm{Hex}}}$.
  • Figure 3: Error constant $M(\theta_1, \delta) = C(\theta_1, \theta_1 + \delta)$ in terms of $\theta_1$ and $\delta = \left(\theta_2 - \theta_1\right)$ (left) and one-dimensional profiles of $M(\theta_1, \delta)$ in terms of $\delta$ for $\theta_1 \in \{0, \frac{\pi}{4}, \frac{4\pi}{3}\}$ (right). In both plots, we only show $M(\theta_1, \delta)$ where $M(\theta_1, \delta) < 10$ for better visual representation.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof