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Graph Signal Inference by Learning Narrowband Spectral Kernels

Osman Furkan Kar, Gülce Turhan, Elif Vural

TL;DR

Graph signals often concentrate energy in multiple spectral bands, not just the lowest frequencies. The authors propose Spectral Graph Kernel Learning (SGKL), a dictionary-based framework that uses Gaussian narrowband kernels in the graph spectral domain, shared across multiple graphs, to jointly learn kernel parameters and sparse signal representations from partially observed data. The approach combines an alternating optimization scheme (ADMM for coefficients and gradient descent for kernel parameters) with a theoretical analysis showing when joint multi-graph learning improves reconstruction, including a bound of $O(1/(M K))$ for unobserved-sample error and a cross-graph threshold $K < O(1/Δ_ψ^2)$. Empirical results on synthetic and real datasets demonstrate competitive signal interpolation accuracy and robustness to missing data and hyperparameter choices, underscoring the method's effectiveness for spectrally concentrated graph signals. These contributions advance data-efficient graph signal modeling beyond band-limited assumptions and enable cross-graph information fusion in practical reconstruction tasks.

Abstract

While a common assumption in graph signal analysis is the smoothness of the signals or the band-limitedness of their spectrum, in many instances the spectrum of real graph data may be concentrated at multiple regions of the spectrum, possibly including mid-to-high-frequency components. In this work, we propose a novel graph signal model where the signal spectrum is represented through the combination of narrowband kernels in the graph frequency domain. We then present an algorithm that jointly learns the model by optimizing the kernel parameters and the signal representation coefficients from a collection of graph signals. Our problem formulation has the flexibility of permitting the incorporation of signals possibly acquired on different graphs into the learning algorithm. We then theoretically study the signal reconstruction performance of the proposed method, by also elaborating on when joint learning on multiple graphs is preferable to learning an individual model on each graph. Experimental results on several graph data sets shows that the proposed method offers quite satisfactory signal interpolation accuracy in comparison with a variety of reference approaches in the literature.

Graph Signal Inference by Learning Narrowband Spectral Kernels

TL;DR

Graph signals often concentrate energy in multiple spectral bands, not just the lowest frequencies. The authors propose Spectral Graph Kernel Learning (SGKL), a dictionary-based framework that uses Gaussian narrowband kernels in the graph spectral domain, shared across multiple graphs, to jointly learn kernel parameters and sparse signal representations from partially observed data. The approach combines an alternating optimization scheme (ADMM for coefficients and gradient descent for kernel parameters) with a theoretical analysis showing when joint multi-graph learning improves reconstruction, including a bound of for unobserved-sample error and a cross-graph threshold . Empirical results on synthetic and real datasets demonstrate competitive signal interpolation accuracy and robustness to missing data and hyperparameter choices, underscoring the method's effectiveness for spectrally concentrated graph signals. These contributions advance data-efficient graph signal modeling beyond band-limited assumptions and enable cross-graph information fusion in practical reconstruction tasks.

Abstract

While a common assumption in graph signal analysis is the smoothness of the signals or the band-limitedness of their spectrum, in many instances the spectrum of real graph data may be concentrated at multiple regions of the spectrum, possibly including mid-to-high-frequency components. In this work, we propose a novel graph signal model where the signal spectrum is represented through the combination of narrowband kernels in the graph frequency domain. We then present an algorithm that jointly learns the model by optimizing the kernel parameters and the signal representation coefficients from a collection of graph signals. Our problem formulation has the flexibility of permitting the incorporation of signals possibly acquired on different graphs into the learning algorithm. We then theoretically study the signal reconstruction performance of the proposed method, by also elaborating on when joint learning on multiple graphs is preferable to learning an individual model on each graph. Experimental results on several graph data sets shows that the proposed method offers quite satisfactory signal interpolation accuracy in comparison with a variety of reference approaches in the literature.

Paper Structure

This paper contains 20 sections, 6 theorems, 110 equations, 6 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Graph Signal Model
  4. Graph Dictionary Model
  5. Graph Signal Model
  6. Proposed Method for Learning Graph Signal Representations
  7. Problem Formulation
  8. Solution of the Optimization Problem
  9. Optimization of the Coefficients ${\mathbf{X}^m}$
  10. Optimization of the Kernel Parameters ${\boldsymbol{\psi}}$
  11. Overall Algorithm
  12. Theoretical Analysis of the Proposed Method
  13. Experimental Results
  14. Performance and Sensitivity Analysis
  15. Effect of noise level
  16. ...and 5 more sections

Key Result

Theorem 1

Let $0<{\tau} <1$ and $0<\varepsilon <1$ be any constants and let ${C_{{R}}^m} \triangleq E[({R^m})^{-2}] \, (2 {N^m} \sigma^2 + 4 (J {N^m} \delta)^2 + \eta_x \eta_w^{-1} J {N^m} \delta )$. Then with probability at least the expected average estimation error of the unobserved samples $\overline{\mathbf{S}}_i^m {\mathbf{y}_i^m}$ of the graph signals ${\mathbf{y}_i^m}$ given by Algorithm alg_SGKL i

Figures (6)

  • Figure 1: (a) A real graph signal consisting of wind speed measurements from the Molène data set Girault-Stationarity (b) The signal spectrum contains middle and high frequencies in addition to low frequencies (blue). Our approach is based on fitting narrowband kernels to high-energy spectral components (red).
  • Figure 2: (a) Variation of the NMSE with the number of signals $K$. (b) The threshold value of $K$ under which joint learning outperforms individual learning.
  • Figure 3: Estimation errors of compared methods on (a), (b): $\mathcal{G}^1$ and (c), (d): $\mathcal{G}^2$ for the synthetic data set. The results of the algorithms with the lowest errors are replotted in right panels (b), (d) for visual clarity.
  • Figure 4: Estimation errors of compared methods on (a), (b): $\mathcal{G}^1$ and (c), (d): $\mathcal{G}^2$ for the Molène data set. The results of the algorithms with the lowest errors are replotted in right panels (b), (d) for visual clarity.
  • Figure 5: Estimation errors of compared methods on (a): $\mathcal{G}^1$ and (b): $\mathcal{G}^2$ for the COVID-19 data set.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • proof
  • ...and 2 more