In this paper, we propose a novel and general framework to construct tight framelet systems on graphs with localized supports based on hierarchical partitions. Our construction provides parametrized graph framelet systems with great generality based on partition trees, by which we are able to find the size of a low-dimensional subspace that best fits the low-rank structure of a family of signals. The orthogonal decomposition of subspaces provides a key ingredient for the definition of "generalized vanishing moments" for graph framelets. In a data-adaptive setting, the graph framelet systems can be learned by solving an optimization problem on Stiefel manifolds with respect to our parameterization. Moreover, such graph framelet systems can be further improved by solving a subsequent optimization problem on Stiefel manifolds, aiming at providing the utmost sparsity for a given family of graph signals. Experimental results show that our learned graph framelet systems perform superiorly in non-linear approximation and denoising tasks.

翻译：本文提出一种新颖且通用的框架，基于层次划分构建具有局部支撑的图紧框架系统。我们的构造通过划分树提供了参数化且高度通用的图框架系统，从而能够找到最佳拟合一组信号低秩结构的低维子空间维度。子空间的正交分解为图框架小波定义"广义消失矩"提供了关键要素。在数据自适应场景下，可通过求解施蒂费尔流形上的优化问题学习图框架系统参数。此外，通过求解后续施蒂费尔流形优化问题，还能进一步提升该类图框架系统性能，旨在为给定图信号族提供最大稀疏性。实验结果表明，所学习的图框架系统在非线性逼近和去噪任务中表现优越。