Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards joint graph learning and sampling set selection from data

Shashank N. Sridhara, Eduardo Pavez, Antonio Ortega

TL;DR

This work proposes Vertex Importance Sampling with Repulsion (VIS), a greedy algorithm where spatially-separated “important” nodes are selected to ensure better reconstruction and shows that sampling using VIS and VISR leads to competitive reconstruction performance and lower complexity than the conventional two-step approach of graph learning followed by graph sampling.

Abstract

We explore the problem of sampling graph signals in scenarios where the graph structure is not predefined and must be inferred from data. In this scenario, existing approaches rely on a two-step process, where a graph is learned first, followed by sampling. More generally, graph learning and graph signal sampling have been studied as two independent problems in the literature. This work provides a foundational step towards jointly optimizing the graph structure and sampling set. Our main contribution, Vertex Importance Sampling (VIS), is to show that the sampling set can be effectively determined from the vertex importance (node weights) obtained from graph learning. We further propose Vertex Importance Sampling with Repulsion (VISR), a greedy algorithm where spatially -separated "important" nodes are selected to ensure better reconstruction. Empirical results on simulated data show that sampling using VIS and VISR leads to competitive reconstruction performance and lower complexity than the conventional two-step approach of graph learning followed by graph sampling.

Towards joint graph learning and sampling set selection from data

TL;DR

This work proposes Vertex Importance Sampling with Repulsion (VIS), a greedy algorithm where spatially-separated “important” nodes are selected to ensure better reconstruction and shows that sampling using VIS and VISR leads to competitive reconstruction performance and lower complexity than the conventional two-step approach of graph learning followed by graph sampling.

Abstract

We explore the problem of sampling graph signals in scenarios where the graph structure is not predefined and must be inferred from data. In this scenario, existing approaches rely on a two-step process, where a graph is learned first, followed by sampling. More generally, graph learning and graph signal sampling have been studied as two independent problems in the literature. This work provides a foundational step towards jointly optimizing the graph structure and sampling set. Our main contribution, Vertex Importance Sampling (VIS), is to show that the sampling set can be effectively determined from the vertex importance (node weights) obtained from graph learning. We further propose Vertex Importance Sampling with Repulsion (VISR), a greedy algorithm where spatially -separated "important" nodes are selected to ensure better reconstruction. Empirical results on simulated data show that sampling using VIS and VISR leads to competitive reconstruction performance and lower complexity than the conventional two-step approach of graph learning followed by graph sampling.

Paper Structure

This paper contains 13 sections, 1 theorem, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and Background
  3. Graph signal processing
  4. Graph and vertex importance learning
  5. Graph Signal Sampling Formulation
  6. Sampling objective
  7. Greedy sampling set selection
  8. Sampling based on vertex importances
  9. Vertex Importance Sampling with Repulsion
  10. Experiments
  11. Comparison between VIS and VISR
  12. Comparison with other sampling algorithms
  13. Conclusion and Future work

Key Result

Theorem 3.1

For small $\mu$, the error covariance can be approximated as where $\gamma = 2\mu - \mu^2$

Figures (3)

  • Figure 1: Vertices selected from the greedy solution to D-optimal sampling objective
  • Figure 2: Visual comparison of vertices selected. (a) learned graph with vertex importance shown in the color bar, (b) sampling set from VIS, (c) sampling set from VISR
  • Figure 3: $\overline{\mathrm{MSE}}$ comparison in different settings. (a) comparison between VIS and VISR with greedy solution to D-optimal sampling objective, (b) comparison with other state-of-the-art sampling algorithms, (c) comparison of using ${\bf L}$ and $({\bf L} + {\bf Q})$ for reconstruction from a randomly selected sampling set.

Theorems & Definitions (2)

  • Theorem 3.1
  • proof