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Introducing Graph Learning over Polytopic Uncertain Graph

Masako Kishida, Shunsuke Ono

TL;DR

The paper addresses graph learning under polytopic uncertainty, where a graph's Laplacian or adjacency can vary within a convex hull of base graphs. By embedding the polytopic constraint into two established frameworks, learning a Laplacian and learning an adjacency matrix, the authors derive convex optimization problems in the simplex of combination weights $\Theta$; these problems reduce the number of free parameters and retain convexity. Numerical experiments on a 20-node random geometric graph with 100 signals show that the polytopic formulations yield competitive or improved metrics such as small $\|\cdot\|_F$ norms and high NMI/F-measures, even when the exact ground truth is not recovered. Overall, the approach offers robust, computationally efficient graph learning under uncertainty and opens avenues for integrating polytopic uncertainty with other graph-learning techniques and applications.

Abstract

This extended abstract introduces a class of graph learning applicable to cases where the underlying graph has polytopic uncertainty, i.e., the graph is not exactly known, but its parameters or properties vary within a known range. By incorporating this assumption that the graph lies in a polytopic set into two established graph learning frameworks, we find that our approach yields better results with less computation.

Introducing Graph Learning over Polytopic Uncertain Graph

TL;DR

The paper addresses graph learning under polytopic uncertainty, where a graph's Laplacian or adjacency can vary within a convex hull of base graphs. By embedding the polytopic constraint into two established frameworks, learning a Laplacian and learning an adjacency matrix, the authors derive convex optimization problems in the simplex of combination weights ; these problems reduce the number of free parameters and retain convexity. Numerical experiments on a 20-node random geometric graph with 100 signals show that the polytopic formulations yield competitive or improved metrics such as small norms and high NMI/F-measures, even when the exact ground truth is not recovered. Overall, the approach offers robust, computationally efficient graph learning under uncertainty and opens avenues for integrating polytopic uncertainty with other graph-learning techniques and applications.

Abstract

This extended abstract introduces a class of graph learning applicable to cases where the underlying graph has polytopic uncertainty, i.e., the graph is not exactly known, but its parameters or properties vary within a known range. By incorporating this assumption that the graph lies in a polytopic set into two established graph learning frameworks, we find that our approach yields better results with less computation.
Paper Structure (7 sections, 1 theorem, 11 equations, 1 figure, 1 table)

This paper contains 7 sections, 1 theorem, 11 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Polytopic uncertain graph
  3. Proposed problem formulations
  4. Learning graph Laplacian
  5. Learning adjacency matrix
  6. Numerical examples
  7. Conclusions

Key Result

Lemma II.1

Any matrix $L\in \mathbf{L}$ is a graph Laplacian if $L_i$, $i = 1, \cdots, p$, are all graph Laplacians.

Figures (1)

  • Figure 1: Matrix heatmaps: 1st row for Section \ref{['sec:dong']}, 2nd row for Section \ref{['sec:kal']}, left for ground truth, middle for DonTF16 and Kal16, right for proposed.

Theorems & Definitions (1)

  • Lemma II.1