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.

翻译：我们研究图上分段光滑信号的估计问题。提出一种基于$\ell_{2,0}$范数惩罚的图趋势滤波（GTF）模型，用于估计节点间具有非均匀光滑程度的分段光滑图信号。我们证明该GTF模型同时实现了节点信号的k均值聚类与图边的最小割分割，且聚类与分割共享相同的分配矩阵。本文提出两种求解该GTF模型的方法：谱分解法与基于模拟退火的方法。在合成数据集与真实数据集上的实验表明，与现有方法相比，所提GTF模型在去噪、支撑恢复与半监督分类任务中均展现出更优性能。此外，对于边集规模较大的数据集，所提模型的求解效率也优于现有方法。