  In deep neural networks, better results can often be obtained by increasing
the complexity of previously developed basic models. However, it is unclear
whether there is a way to boost performance by decreasing the complexity of
such models. Here, based on an optimization method, we investigate the
feasibility of improving graph classification performance while simplifying the
model learning process. Inspired by progress in structural information
assessment, we optimize the given data sample from graphs to encoding trees. In
particular, we minimize the structural entropy of the transformed encoding tree
to decode the key structure underlying a graph. This transformation is denoted
as structural optimization. Furthermore, we propose a novel feature combination
scheme, termed hierarchical reporting, for encoding trees. In this scheme,
features are transferred from leaf nodes to root nodes by following the
hierarchical structures of encoding trees. We then present an implementation of
the scheme in a tree kernel and a convolutional network to perform graph
classification. The tree kernel follows label propagation in the
Weisfeiler-Lehman (WL) subtree kernel, but it has a lower runtime complexity
$O(n)$. The convolutional network is a special implementation of our tree
kernel in the deep learning field and is called Encoding Tree Learning (ETL).
We empirically validate our tree kernel and convolutional network with several
graph classification benchmarks and demonstrate that our methods achieve better
performance and lower computational consumption than competing approaches.
