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Uncertainty Principle for Vertex-Time Graph Signal Processing

Yanan Zhao, Xingchao Jian, Feng Ji, Wee Peng Tay, Antonio Ortega

TL;DR

This work introduces an uncertainty principle for vertex-time graph signals that unifies classical time-frequency and graph localization within a joint vertex-time–spectral-frequency framework. It defines projection-based spreads and shows how maximally concentrated atoms can form a vertex-time dictionary for robust reconstruction under intermittent observations, alongside a continuous-time perspective that handles asynchronous sampling. The authors propose two main contributions: (i) JECD, a structured, energy-concentrated dictionary (with vertex-subset and continuous-parameter optimization, plus a Gaussian-subspace variant), and (ii) ECGL, a graph-learning approach that maximizes spectral-energy concentration while enforcing VT localization trade-offs. Experimental results on synthetic and real data demonstrate improved reconstruction accuracy, resilience to noise, and superior graph topology inference compared to existing methods, highlighting practical impact for sensor and social networks with irregular data collection.

Abstract

We present an uncertainty principle for graph signals in the vertex-time domain, unifying the classical time-frequency and graph uncertainty principles within a single framework. By defining vertex-time and spectral-frequency spreads, we quantify signal localization across these domains. Our framework identifies a class of signals that achieve maximum concentration in both the spatial and temporal domains. These signals serve as fundamental atoms for a new vertex-time dictionary, enhancing signal reconstruction under practical constraints, such as intermittent data commonly encountered in sensor and social networks. Furthermore, we introduce a novel graph topology inference method leveraging the uncertainty principle. Numerical experiments on synthetic and real datasets validate the effectiveness of our approach, demonstrating improved reconstruction accuracy, greater robustness to noise, and enhanced graph learning performance compared to existing methods.

Uncertainty Principle for Vertex-Time Graph Signal Processing

TL;DR

This work introduces an uncertainty principle for vertex-time graph signals that unifies classical time-frequency and graph localization within a joint vertex-time–spectral-frequency framework. It defines projection-based spreads and shows how maximally concentrated atoms can form a vertex-time dictionary for robust reconstruction under intermittent observations, alongside a continuous-time perspective that handles asynchronous sampling. The authors propose two main contributions: (i) JECD, a structured, energy-concentrated dictionary (with vertex-subset and continuous-parameter optimization, plus a Gaussian-subspace variant), and (ii) ECGL, a graph-learning approach that maximizes spectral-energy concentration while enforcing VT localization trade-offs. Experimental results on synthetic and real data demonstrate improved reconstruction accuracy, resilience to noise, and superior graph topology inference compared to existing methods, highlighting practical impact for sensor and social networks with irregular data collection.

Abstract

We present an uncertainty principle for graph signals in the vertex-time domain, unifying the classical time-frequency and graph uncertainty principles within a single framework. By defining vertex-time and spectral-frequency spreads, we quantify signal localization across these domains. Our framework identifies a class of signals that achieve maximum concentration in both the spatial and temporal domains. These signals serve as fundamental atoms for a new vertex-time dictionary, enhancing signal reconstruction under practical constraints, such as intermittent data commonly encountered in sensor and social networks. Furthermore, we introduce a novel graph topology inference method leveraging the uncertainty principle. Numerical experiments on synthetic and real datasets validate the effectiveness of our approach, demonstrating improved reconstruction accuracy, greater robustness to noise, and enhanced graph learning performance compared to existing methods.
Paper Structure (24 sections, 79 equations, 5 figures, 1 table)

This paper contains 24 sections, 79 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Uncertainty Principle for Vertex-time Graph Signals
  3. Fourier transforms
  4. Vertex-time and spectral-frequency spreads
  5. Uncertainty principle for vertex-time graph signals
  6. Special case of vertex-time and spectral-frequency subsets with product structure
  7. Vertex-time graph signal reconstruction
  8. Localization properties
  9. Joint parametric dictionary learning with support-limited subspace for vertex-time signal reconstruction
  10. Vertex subset selection
  11. Optimization of continuous dictionary parameters $(t_{c},\ell)$
  12. Gaussian subspace basis for vertex-time graph signal reconstruction
  13. Graph topology inference using vertex-time samples
  14. Experimental Results
  15. Vertex-time graph signal reconstruction
  16. ...and 9 more sections

Figures (5)

  • Figure 1: The number of reported cases across counties and days from July 29, 2020, to August 1, 2022, for the Covid-19 dataset.
  • Figure 2: Feasible region $\Theta$ of $\alpha_{\mathop{\mathrm{VT}}\nolimits}$ and $\beta_{\mathop{\mathrm{SF}}\nolimits}$
  • Figure 3: Reconstruction error over test set under different conditions. Each point in the figure is obtained by 10 repetitions.
  • Figure 4: Reconstruction error under varying conditions. Each data point is the average of 10 experiments.
  • Figure 5: Ground-truth graph siganl and reconstruction error under varying SNR levels.

Theorems & Definitions (2)

  • proof
  • proof