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Graph Canonical Coherence Analysis

Kyusoon Kim, Hee-Seok Oh

TL;DR

Graph canonical coherence analysis (gCChA) extends canonical correlation analysis to multivariate graph signals in the graph frequency domain, addressing the irregularity of graphs by operating on spectral components. The method constructs low-dimensional canonical graph signals through graph-filtered representations and seeks maximal coherence across graph frequencies, yielding interpretable, multiscale relationships via population and sample graph signals. Theoretical results (e.g., A(λ)=G(λ)H(λ) and γ_i(λ) via eigenvalues τ_i(λ)) underpin a practical framework with loadings, cross-loadings, and adequacy measures to interpret how original signals project onto canonical graph directions. Empirically, gCChA demonstrates meaningful spectral patterns and improved classification performance on G20 economic-energy data and USPS digits relative to vertex-domain approaches, highlighting its potential for analyzing cross-graph dependencies in networked data.

Abstract

We propose graph canonical coherence analysis (gCChA), a novel framework that extends canonical correlation analysis to multivariate graph signals in the graph frequency domain. The proposed method addresses challenges posed by the inherent features of graphs: discreteness, finiteness, and irregularity. It identifies pairs of canonical graph signals that maximize their coherence, enabling the exploration of relationships between two sets of graph signals from a spectral perspective. This framework shows how these relationships change across different structural scales of the graph. We demonstrate the usefulness of this method through applications to economic and energy datasets of G20 countries and the USPS handwritten digit dataset.

Graph Canonical Coherence Analysis

TL;DR

Graph canonical coherence analysis (gCChA) extends canonical correlation analysis to multivariate graph signals in the graph frequency domain, addressing the irregularity of graphs by operating on spectral components. The method constructs low-dimensional canonical graph signals through graph-filtered representations and seeks maximal coherence across graph frequencies, yielding interpretable, multiscale relationships via population and sample graph signals. Theoretical results (e.g., A(λ)=G(λ)H(λ) and γ_i(λ) via eigenvalues τ_i(λ)) underpin a practical framework with loadings, cross-loadings, and adequacy measures to interpret how original signals project onto canonical graph directions. Empirically, gCChA demonstrates meaningful spectral patterns and improved classification performance on G20 economic-energy data and USPS digits relative to vertex-domain approaches, highlighting its potential for analyzing cross-graph dependencies in networked data.

Abstract

We propose graph canonical coherence analysis (gCChA), a novel framework that extends canonical correlation analysis to multivariate graph signals in the graph frequency domain. The proposed method addresses challenges posed by the inherent features of graphs: discreteness, finiteness, and irregularity. It identifies pairs of canonical graph signals that maximize their coherence, enabling the exploration of relationships between two sets of graph signals from a spectral perspective. This framework shows how these relationships change across different structural scales of the graph. We demonstrate the usefulness of this method through applications to economic and energy datasets of G20 countries and the USPS handwritten digit dataset.
Paper Structure (17 sections, 5 theorems, 34 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 5 theorems, 34 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Graph Signals and Graph Frequency Domain
  4. Stationarity and Spectral Densities on Graphs
  5. Related Work
  6. Proposed Method
  7. Graph Canonical Coherence Analysis and Its Properties
  8. Interpretation of Graph Canonical Coherence Analysis
  9. Numerical Experiments
  10. G20 Network Data Analysis
  11. Classification Analysis of USPS Dataset
  12. Concluding Remarks
  13. Proofs
  14. Proof of Theorem 3.1
  15. Proof of Theorem 3.2
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $XY=(X_1 \mid \cdots \mid X_p \mid Y_1 \mid \cdots \mid Y_q)$ be a $(p+q)$-dimensional stationary graph signal with respect to the graph shift operator $S=V\Lambda V^H$ on $\mathcal{G}$. The means of $X_i$ and $Y_j$ are denoted by $\mu_i^X$ and $\mu_j^Y$, respectively. Let $p_{ij}^X$ and $p_{ij} whose $(i,j)$th elements are $p_{ij}^X(\lambda_\ell)$, $p_{ij}^Y(\lambda_\ell)$, $p_{ij}^{XY}(\lamb

Figures (6)

  • Figure 1: Illustration of univariate (left) and multivariate (right) graph signals.
  • Figure 2: Trading network among G20 countries in 2019, where darker edges represent stronger trade connections.
  • Figure 3: Graph canonical coherences $\hat{\gamma}_i(\lambda_\ell)$ for the canonical graph signal pairs $(\hat{Z}_i, \hat{W}_i)$, $i=1,\ldots,5$, across graph frequencies.
  • Figure 4: Basis signals $v_1$ and $v_7$.
  • Figure 5: Graph canonical loadings of the first two canonical graph signals at graph frequencies $\lambda_1$ (top row) and $\lambda_7$ (bottom row). Red dots show positive correlations in the frequency domain, while blue dots represent negative correlations. Loadings with a magnitude exceeding 0.2 (shown by the dashed line) are used for interpretation.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.1
  • ...and 5 more