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Locally supported, quasi-interpolatory bases for the approximation of functions on graphs

Edward J. Fuselier, John Paul Ward

TL;DR

This work proposes perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines, and considers basis functions that have local support.

Abstract

Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present $\ell_\infty$ error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$.

Locally supported, quasi-interpolatory bases for the approximation of functions on graphs

TL;DR

This work proposes perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines, and considers basis functions that have local support.

Abstract

Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate .

Paper Structure

This paper contains 16 sections, 7 theorems, 82 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The Graph Laplacian and Variants
  4. Weighted Spaces and the Connection Between $\mathcal{L}_{rw}$ and $\mathcal{L}_{n}$
  5. The Dirichlet Laplacian
  6. Polyharmonic Splines on Graphs and Lagrange Functions
  7. Function and Laplacian indexing and properties
  8. Assumptions on $\mathcal{G}$
  9. Relating Lagrange and local Lagrange functions
  10. Lagrange Comparison for the Random Walk Laplacian
  11. Lagrange Comparison for the Standard Laplacian
  12. Experiments
  13. Convergence of bases and error bounds
  14. Computation time
  15. Connectivity of Unknowns
  16. ...and 1 more sections

Key Result

Lemma 3.1

If $L = \mathcal{L}_{rw}$ and the graph satisfies Assumption as:unknowndegree, the matrix satisfies where $d_m:=\min_{x\in U}d_x$.

Figures (7)

  • Figure 1: Given a center $v$, we will choose some subset $\Omega\subset\mathcal{V}$ containing $v$ on which to compute the local Lagrange functions. In this example nodes within $\Omega$ are unshaded. Unshaded circles are in $\hbox{int}\left(\Omega\right)$, unshaded squares comprise $\delta_i\left(\Omega\right)$, and the shaded circles make up $\delta_o\left(\Omega\right)$. Interestingly, note that it is possible to have points in the interior of $\Omega$ that are farther away from the center than points on the inner boundary.
  • Figure 2: Graph generated from the Fibonacci lattice on the sphere with 125 points/vertices.
  • Figure 3: Comparison of a Lagrange and local Lagrange basis vector for $10^4$ (solid red), $10^5$ (dashed blue), $10^6$ (green dots) points on the sphere. The horizontal axis indicates the number of unknown points $n_U$ in the neighborhood used to define the local Lagrange function.
  • Figure 4: Illustration of the error bound using a graph on the sphere with $5,000$ vertices. The bottom horizontal axis denotes the radius $R$ of the footprint $\Omega$. The top horizontal axis indicates $n_U$, the number of unknown vertices inside $\Omega$. Dash Blue line: error bound; Black line: Difference between Lagrange and Local Lagrange basis functions.
  • Figure 5: Computation time comparison between Lagrange and local Lagrange. Red dashed line: Lagrange basis, simultaneous computation in dense format; Black dot-dashed line: Lagrange basis, serial computation in sparse format; Solid blue line: Local Lagrange basis, serial computation. Green triangles: Local Lagrange basis, parallel implementation.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Proposition A.1
  • ...and 2 more