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Inhomogeneous graph trend filtering via a l2,0 cardinality penalty

Xiaoqing Huang, Andersen Ang, Kun Huang, Jie Zhang, Yijie Wang

TL;DR

This work addresses estimating piecewise smooth signals on graphs when smoothness is inhomogeneous by introducing a first-order Graph Trend Filtering model with an $\ell_{2,0}$-norm penalty. The authors prove that the model performs simultaneous $k$-means clustering on node signals and a minimum graph cut on the graph, sharing the same assignment matrix, and propose two solving strategies: a spectral approximation and a heat-bath simulated annealing method. Empirical results on synthetic and real datasets show improved denoising, support recovery, and semi-supervised classification, with favorable scalability on large edge sets. The paper also extends the framework to graph-based transductive learning and provides theoretical characterizations (Theorems 0–1) to justify the coupling of data fidelity and topology, demonstrating practical impact for graph-based signal processing tasks.

Abstract

We study estimation of piecewise smooth signals over a graph. We propose a $\ell_{2,0}$-norm penalized Graph Trend Filtering (GTF) model to estimate piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness across the nodes. We prove that the proposed GTF model is simultaneously a k-means clustering on the signal over the nodes and a minimum graph cut on the edges of the graph, where the clustering and the cut share the same assignment matrix. We propose two methods to solve the proposed GTF model: a spectral decomposition method and a method based on simulated annealing. In the experiment on synthetic and real-world datasets, we show that the proposed GTF model has a better performances compared with existing approaches on the tasks of denoising, support recovery and semi-supervised classification. We also show that the proposed GTF model can be solved more efficiently than existing models for the dataset with a large edge set.

Inhomogeneous graph trend filtering via a l2,0 cardinality penalty

TL;DR

This work addresses estimating piecewise smooth signals on graphs when smoothness is inhomogeneous by introducing a first-order Graph Trend Filtering model with an -norm penalty. The authors prove that the model performs simultaneous -means clustering on node signals and a minimum graph cut on the graph, sharing the same assignment matrix, and propose two solving strategies: a spectral approximation and a heat-bath simulated annealing method. Empirical results on synthetic and real datasets show improved denoising, support recovery, and semi-supervised classification, with favorable scalability on large edge sets. The paper also extends the framework to graph-based transductive learning and provides theoretical characterizations (Theorems 0–1) to justify the coupling of data fidelity and topology, demonstrating practical impact for graph-based signal processing tasks.

Abstract

We study estimation of piecewise smooth signals over a graph. We propose a -norm penalized Graph Trend Filtering (GTF) model to estimate piecewise smooth graph signals that exhibit inhomogeneous levels of smoothness across the nodes. We prove that the proposed GTF model is simultaneously a k-means clustering on the signal over the nodes and a minimum graph cut on the edges of the graph, where the clustering and the cut share the same assignment matrix. We propose two methods to solve the proposed GTF model: a spectral decomposition method and a method based on simulated annealing. In the experiment on synthetic and real-world datasets, we show that the proposed GTF model has a better performances compared with existing approaches on the tasks of denoising, support recovery and semi-supervised classification. We also show that the proposed GTF model can be solved more efficiently than existing models for the dataset with a large edge set.
Paper Structure (44 sections, 5 theorems, 29 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 44 sections, 5 theorems, 29 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. GTF for scalar-valued graph signals
  3. GTF with the counting norm penalty
  4. The advantage of using the proposed norm penalty
  5. GTF for vector-valued graph signals
  6. Contributions and paper organization
  7. Equivalent formulation of
  8. Theoretical characterizations of the proposed GTF model
  9. Simultaneous k-means and graph cut
  10. The algorithms to solve $\mathcal{P} _1$
  11. Spectral approximation
  12. Eliminate
  13. is approximately a vector partition problem
  14. Approximation error and the choice of $\alpha$
  15. The whole approximate spectral method
  16. ...and 29 more sections

Key Result

Theorem 1

Given a data matrix $\bm{B} \in \mathbb{R}^{n \times d}$ over the nodes $V$ of a simple connected unweighted undirected graph graph $G(V,E)$, if $\mathcal{L}(\bm{B}) > 0$, then it indicates the existence of an edge subset that cuts $G$ into $k$ disjoint partitions $C=\{C_1, \dots, C_k\}$. Furthermore, $\mathcal{L}(\bm{B}) > 0$ enforces the node signals in those $k$ partitions sharing the same val

Figures (3)

  • Figure 1: Toy examples. (a) The ground truth of an inhomogeneous scalar-valued graph signal. (b) The same graph signal in (a) with added Gaussian noise $\mathcal{N}(0,0.1)$. (c) Comparison of the recovery of the ground truth in (a) from 20 different noisy inhomogeneous graph signals as in (b) with different parameter $\lambda$ in \ref{['prob:0']} and \ref{['l0model']}. The proposed model \ref{['l0model']} achieves the lowest mean square error when screening the parameter $\lambda$. (d) An example of a vector-valued graph signal.
  • Figure 2: On identifying boundary edges. Left: the ground truth $\bm{y}_*$ on the Minnesota road graph. Middle: the ROC curve of the methods. The curves for the spectral method and SA method are overlapped. Here the $\ell_0$ model achieved perfect ROC. Right: the optimal $k_*$ ($=4$ clusters) for the $\ell_0$ model $\mathcal{P} _2$.
  • Figure 3: Left: the graph structure of $G_1$ (top) and $G_2$ (bottom) and the corresponding ground truth signal input to each node in both graphs. Right: the corresponding plots of input signal SNR versus reconstructed signal SNR for the GTF models. Note that in both graphs, the $\ell_{2,0}$ model consistently outperforms other GFT models.

Theorems & Definitions (9)

  • Theorem 1: The effect of $\mathcal{L}(\bm{B}) > 0$ in $\mathcal{P} _0$
  • Theorem 2: $\mathcal{P} _0$ = k-means and min-cut
  • proof
  • Lemma 3: Simplifying $\mathcal{P} _1$
  • proof
  • Theorem 4: Vector partition
  • proof
  • Theorem 5
  • proof