  Graph representation learning has many real-world applications, from
super-resolution imaging, 3D computer vision to drug repurposing, protein
classification, social networks analysis. An adequate representation of graph
data is vital to the learning performance of a statistical or machine learning
model for graph-structured data. In this paper, we propose a novel multiscale
representation system for graph data, called decimated framelets, which form a
localized tight frame on the graph. The decimated framelet system allows
storage of the graph data representation on a coarse-grained chain and
processes the graph data at multi scales where at each scale, the data is
stored at a subgraph. Based on this, we then establish decimated G-framelet
transforms for the decomposition and reconstruction of the graph data at multi
resolutions via a constructive data-driven filter bank. The graph framelets are
built on a chain-based orthonormal basis that supports fast graph Fourier
transforms. From this, we give a fast algorithm for the decimated G-framelet
transforms, or FGT, that has linear computational complexity O(N) for a graph
of size N. The theory of decimated framelets and FGT is verified with numerical
examples for random graphs. The effectiveness is demonstrated by real-world
applications, including multiresolution analysis for traffic network, and graph
neural networks for graph classification tasks.
