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：本研究针对图上的分段平滑信号估计问题展开。我们提出了一种基于ℓ₂,₀范数惩罚的图趋势滤波（GTF）模型，用于估计在节点间呈现非均匀平滑度的分段平滑图信号。我们证明了所提出的GTF模型同时等价于信号在节点上的k均值聚类与图边上的最小割问题，且该聚类与割共享相同的分配矩阵。针对该GTF模型，我们提出了两种求解方法：谱分解方法与基于模拟退火的方法。在合成数据集与真实数据集的实验中，我们证明所提出的GTF模型在去噪、支撑集恢复和半监督分类任务上均优于现有方法。我们还表明，对于具有大规模边集的数据集，所提出的GTF模型比现有模型具有更高的求解效率。