Inhomogeneous Graph Trend Filtering via a $\ell _{2,0}$-Norm Cardinality Penalty

We study estimation of piecewise smooth signals over a graph. We propose a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub><mml:mrow><mml:mo>ℓ</mml:mo></mml:mrow> <mml:mrow><mml:mn>2,0</mml:mn></mml:mrow> </mml:msub> </mml:math> -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.