  Graph convolutional networks (GCNs) are widely used in graph-based
applications such as graph classification and segmentation. However, current
GCNs have limitations on implementation such as network architectures due to
their irregular inputs. In contrast, convolutional neural networks (CNNs) are
capable of extracting rich features from large-scale input data, but they do
not support general graph inputs. To bridge the gap between GCNs and CNNs, in
this paper we study the problem of how to effectively and efficiently map
general graphs to 2D grids that CNNs can be directly applied to, while
preserving graph topology as much as possible. We therefore propose two novel
graph-to-grid mapping schemes, namely, {\em graph-preserving grid layout
(GPGL)} and its extension {\em Hierarchical GPGL (H-GPGL)} for computational
efficiency. We formulate the GPGL problem as integer programming and further
propose an approximate yet efficient solver based on a penalized Kamada-Kawai
method, a well-known optimization algorithm in 2D graph drawing. We propose a
novel vertex separation penalty that encourages graph vertices to lay on the
grid without any overlap. Along with this image representation, even extra 2D
maxpooling layers contribute to the PointNet, a widely applied point-based
neural network. We demonstrate the empirical success of GPGL on general graph
classification with small graphs and H-GPGL on 3D point cloud segmentation with
large graphs, based on 2D CNNs including VGG16, ResNet50 and multi-scale maxout
(MSM) CNN.
