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In this paper, we propose a novel and general framework to construct tightframelet systems on graphs with localized supports based on hierarchicalpartitions. Our construction provides parametrized graph framelet systems withgreat generality based on partition trees, by which we are able to find thesize of a low-dimensional subspace that best fits the low-rank structure of afamily of signals. The orthogonal decomposition of subspaces provides a keyingredient for the definition of "generalized vanishing moments" for graphframelets. In a data-adaptive setting, the graph framelet systems can belearned by solving an optimization problem on Stiefel manifolds with respect toour parameterization. Moreover, such graph framelet systems can be furtherimproved 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 performsuperiorly in non-linear approximation and denoising tasks.

Data-Adaptive Graph Framelets with Generalized Vanishing Moments for Graph Signal Processing