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Learning Laplacian Forms for Graph Signal Processing via the Deformed Laplacian

Stefania Sardellitti

Abstract

Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been considered depending on the type of features to be extracted from the data. The main contribution of this paper is the introduction of a parametric Laplacian, called the deformed Laplacian, defined as a quadratic matrix polynomial that provides a parametric dictionary for graph signal processing. The deformed Laplacian can be interpreted as the generator of a parametric linear reaction-diffusion dynamics on graphs, capturing the interplay between diffusive coupling and nodal reaction effects. It is a parametric polynomial matrix that enables the design of novel topological operators tailored to both the underlying graph structure and the observed signals. Interestingly, we show that several Laplacian variants proposed in the literature arise as special cases of the deformed Laplacian. We then develop a method to jointly learn the deformed Laplacian and the graph signals from data, showing how its use improves signal representation across a broad class of graphs compared to standard Laplacian forms. Through extensive numerical experiments on both synthetic and real-world datasets, including financial and communication networks, we assess the benefits of the proposed method in terms of graph signal reconstruction error and sparsity of the representation.

Learning Laplacian Forms for Graph Signal Processing via the Deformed Laplacian

Abstract

Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been considered depending on the type of features to be extracted from the data. The main contribution of this paper is the introduction of a parametric Laplacian, called the deformed Laplacian, defined as a quadratic matrix polynomial that provides a parametric dictionary for graph signal processing. The deformed Laplacian can be interpreted as the generator of a parametric linear reaction-diffusion dynamics on graphs, capturing the interplay between diffusive coupling and nodal reaction effects. It is a parametric polynomial matrix that enables the design of novel topological operators tailored to both the underlying graph structure and the observed signals. Interestingly, we show that several Laplacian variants proposed in the literature arise as special cases of the deformed Laplacian. We then develop a method to jointly learn the deformed Laplacian and the graph signals from data, showing how its use improves signal representation across a broad class of graphs compared to standard Laplacian forms. Through extensive numerical experiments on both synthetic and real-world datasets, including financial and communication networks, we assess the benefits of the proposed method in terms of graph signal reconstruction error and sparsity of the representation.

Paper Structure

This paper contains 14 sections, 4 theorems, 30 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Matrix Polynomials: A brief overview
  3. The Deformed Laplacian
  4. The Deformed Laplacian as a Reaction-Diffusion Operator
  5. The Deformed Graph Laplacian: A Unified Laplacian form
  6. The combinatorial graph Laplacian
  7. The signless graph Laplacian
  8. The Signed Laplacian
  9. Spectral properties of the Deformed Laplacian
  10. Spectral Signal representation via deformed graph Laplacians
  11. Joint learning of Laplacian form and graph signals
  12. Synthetic Data Experiments
  13. Real Data Experiments
  14. Conclusions

Key Result

Proposition 1

For any graph, the following statements hold: $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Second eigenvector of $\hbox{$\mathbf{L}$}$ on the Zachary's karate club network.
  • Figure 2: Eigenvector of $\hbox{$\mathbf{L}$}_{\text{SL}}$ associate with the zero eigenvalue on the Southern Women bipartite network.
  • Figure 3: First eigenvector of $\hbox{$\mathbf{L}$}_{\text{S}}$ associate with the zero eigenvalue.
  • Figure 4: Node signal on a quasi-bipartite, clustered graph.
  • Figure 5: Signal recovering error versus the parameter $r$ with $K=4$.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4