In this work, we proposed a novel and general method to construct tight frames on graphs with compact supports based on a series of hierarchical partitions. Starting from our abstract construction that generalizes previous methods based on partition trees, we are able to flexibly incorporate subgraph Laplacians into our design of graph frames. Consequently, our general methods permit adjusting the (subgraph) vanishing moments of the framelets and extra properties, such as directionality, for efficiently representing graph signals with path-like supports. Several variants are explicitly defined and tested. Experimental results show our proposed graph frames perform superiorly in non-linear approximation tasks.

翻译：本文提出了一种新颖且通用的方法，基于一系列层次化划分构造具有紧支撑的图紧框架。从我们的抽象构造出发——该构造推广了先前基于划分树的方法——我们能够灵活地将子图拉普拉斯算子融入图框架的设计中。因此，我们的通用方法允许调整（子图）框架的小波消失矩及额外性质（如方向性），以有效表示具有路径状支撑的图信号。文中明确定义了多种变体并进行了测试。实验结果表明，我们提出的图框架在非线性逼近任务中表现出优越性能。