  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.
