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Compressive Recovery of Signals Defined on Perturbed Graphs

Sabyasachi Ghosh, Ajit Rajwade

TL;DR

This work tackles compressive recovery of graph signals when the underlying graph is only approximately known due to small edge perturbations. It introduces CPGR and two algorithms, Ges and Ilecir, which jointly recover the signal and refine the graph using cross-validated compressed sensing. Theoretical recovery guarantees are provided for the brute-force variant, with practical speedups via approximate eigendecompositions. Empirical results on synthetic graphs and image patches demonstrate significant improvements over nominal-graph baselines, with notable edge-preserving benefits and robustness to noise. The framework is extensible to alternative graph-based regularizers, such as Graph Total Variation, broadening its applicability to diverse graph-signal scenarios.

Abstract

Recovery of signals with elements defined on the nodes of a graph, from compressive measurements is an important problem, which can arise in various domains such as sensor networks, image reconstruction and group testing. In some scenarios, the graph may not be accurately known, and there may exist a few edge additions or deletions relative to a ground truth graph. Such perturbations, even if small in number, significantly affect the Graph Fourier Transform (GFT). This impedes recovery of signals which may have sparse representations in the GFT bases of the ground truth graph. We present an algorithm which simultaneously recovers the signal from the compressive measurements and also corrects the graph perturbations. We analyze some important theoretical properties of the algorithm. Our approach to correction for graph perturbations is based on model selection techniques such as cross-validation in compressed sensing. We validate our algorithm on signals which have a sparse representation in the GFT bases of many commonly used graphs in the network science literature. An application to compressive image reconstruction is also presented, where graph perturbations are modeled as undesirable graph edges linking pixels with significant intensity difference. In all experiments, our algorithm clearly outperforms baseline techniques which either ignore the perturbations or use first order approximations to the perturbations in the GFT bases.

Compressive Recovery of Signals Defined on Perturbed Graphs

TL;DR

This work tackles compressive recovery of graph signals when the underlying graph is only approximately known due to small edge perturbations. It introduces CPGR and two algorithms, Ges and Ilecir, which jointly recover the signal and refine the graph using cross-validated compressed sensing. Theoretical recovery guarantees are provided for the brute-force variant, with practical speedups via approximate eigendecompositions. Empirical results on synthetic graphs and image patches demonstrate significant improvements over nominal-graph baselines, with notable edge-preserving benefits and robustness to noise. The framework is extensible to alternative graph-based regularizers, such as Graph Total Variation, broadening its applicability to diverse graph-signal scenarios.

Abstract

Recovery of signals with elements defined on the nodes of a graph, from compressive measurements is an important problem, which can arise in various domains such as sensor networks, image reconstruction and group testing. In some scenarios, the graph may not be accurately known, and there may exist a few edge additions or deletions relative to a ground truth graph. Such perturbations, even if small in number, significantly affect the Graph Fourier Transform (GFT). This impedes recovery of signals which may have sparse representations in the GFT bases of the ground truth graph. We present an algorithm which simultaneously recovers the signal from the compressive measurements and also corrects the graph perturbations. We analyze some important theoretical properties of the algorithm. Our approach to correction for graph perturbations is based on model selection techniques such as cross-validation in compressed sensing. We validate our algorithm on signals which have a sparse representation in the GFT bases of many commonly used graphs in the network science literature. An application to compressive image reconstruction is also presented, where graph perturbations are modeled as undesirable graph edges linking pixels with significant intensity difference. In all experiments, our algorithm clearly outperforms baseline techniques which either ignore the perturbations or use first order approximations to the perturbations in the GFT bases.
Paper Structure (18 sections, 5 theorems, 43 equations, 15 figures, 2 algorithms)

This paper contains 18 sections, 5 theorems, 43 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Related Work
  4. Method
  5. Recovery using standard CS decoding algorithms
  6. Greedy Edge Selection
  7. Inferred Linear-Edge Compressive Image Recovery
  8. Recovery Guarantees and Bounds
  9. Alternatives to Eigendecomposition for GFT basis computation
  10. Empirical Evaluation
  11. Greedy Edge Selection on synthetic graphs
  12. Inferred Linear-Edge Compressive Image Recovery
  13. Conclusion and Future Work
  14. Statement and Proof of Key Theorem
  15. Details Regarding Random Graph Models
  16. ...and 3 more sections

Key Result

Theorem 1

If in the Bfgs algorithm in Sec. subsec:greedy_edge_selection, the number of CV measurements $m_\text{cv}$ obeys for arbitrary constants $c \in (1, \infty)$ and $\delta \in (0, 1)$, then its recovery error is bounded as with probability more than $1-\delta$. As a consequence, if for all graphs $\mathcal{G} \neq \mathcal{G}_{\text{actual}}$ which are upto $d_0$ edge perturbations from $\mathcal{

Figures (15)

  • Figure 1: \ref{['fig:lattice_graph']} A $5\times 5$ 2-D lattice graph whose nodes represent pixels of a $5\times 5$ patch and whose edges represent connections between a pixels and its four neighbors. This forms the nominal graph for the problem of compressive image patch recovery. \ref{['fig:lattice_graph_partitioned']} The lattice graph partitioned by an image edge (orange line). The graph edges going across the image edge are removed (dotted purple lines). Since the image edge is unknown before reconstruction, the image-edge partitioned graph is the (unknown) actual graph for the problem of recovery of an image patch from compressive measurements. \ref{['fig:dct_basis']} All 64 2-D DCT basis vectors of an $8\times 8$ patch. \ref{['fig:patch']} An $8\times 8$ patch with a sharp edge. \ref{['fig:seg_aware_basis']} Segmentation-aware basis vectors for this patch, obtained by computing the eigenvectors of the Laplacian matrix of the graph created by dropping the edges of the $8\times8$ lattice graph whose endpoints lie in different segments of the patch.
  • Figure 2: RRMSE of signal recovered from noiseless measurements ($m = 50, n = 100$ except for KCG where $m = 17, n = 34$) via Greedy Edge Selection (Ges), Lasso with the nominal graph (Ngft-Lasso-Cv), and Lasso with the actual graph (Agft-Lasso-Cv), for various number of perturbed edges. Some bars are not visible due to the value being close to zero.
  • Figure 3: Fraction of cases in which actual graph was recovered by Ges, for various number of perturbed edges, for $m = 50, n = 100$ except for KCG where $m = 17, n = 34$.
  • Figure 4: The average number of edge perturbations successfully detected and the average number of spurious edge perturbations reported by Ges, for various number of perturbed edges on different graph models ($m = 50, n = 100$ except for KCG where $m = 17, n = 34$).
  • Figure 5: Performance of Ges with measurement noise of different levels given by $\beta \in \{0.01,0.02,0.05\}$ ($m = 50, n = 100$).
  • ...and 10 more figures

Theorems & Definitions (5)

  • Theorem 1: Brute-Force Algorithm Recovery Guarantee
  • Theorem 2: Greedy Edge Selection Solution Improvement Guarantee
  • Theorem 3: Ilecir Solution Improvement Guarantee
  • Theorem 4: Brute-Force Algorithm Recovery Guarantee, Theorem \ref{['thm:brute_force_guarantee']} of main paper
  • Theorem 5