🤖 AI Summary
We address the problem of estimating piecewise-smooth graph signals exhibiting heterogeneous local smoothness. To this end, we propose a graph trend filtering model regularized by the ℓ₂,₀ norm. This is the first formulation proven to be equivalent to the joint optimization of k-means clustering with a shared assignment matrix and minimum graph cut—thereby unifying non-uniform smoothing capability, combinatorial interpretability of the solution structure, and an explicit piecewise-constant prior. Our method leverages spectral decomposition for acceleration and employs simulated annealing for optimization. Empirically, it significantly outperforms state-of-the-art graph filtering and sparse regularization approaches on denoising, support recovery, and semi-supervised classification tasks. Moreover, it achieves superior computational efficiency on large-scale sparse graphs.
📝 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.