Our goal is to compare the efficiency of so-called Geometric Convolutions, which can be defined on an arbitrary mesh structure, to standard convolutions, which assume a that the nodes are organised along an Euclidean grid.

These scripts propose several implementations of geometric convolution methods that can be used for Convolutional Neural Network. They are inspired from original paper and codes of [gcn] and [MoNet].

In each scenario, we assume that we are doing convolutions over a graph which is organised as a grid where each node has 8 neighbours. Convolutions are done in a CNN setting with same number of input and output features.

To run the code, use:
python BenchmarkStandardConvolution 1 50 50 where the paramters correspond to Batch Size, grid size (one side of the square), number of features.

nb: You may have to use:

export LD_LIBRARY_PATH=/usr/local/cuda/extras/CUPTI/lib64:$LD_LIBRARY_PATH

to be able to call the timeline library which is used for TensorFlow code profiling.

We use Nvidia Pascal Titan X GPU for all GPU computations. The code was written for Python2.7 and TensorFlow-1.2

The 4 scripts that we provide correspond to the following setting:

Average Time 0.00098 s. It runs on the GPU such a speed is possible thanks to the regular grid organisation.

Average Time 0.00889 s. TensorFlow takes advantage of the pre-computed Sparse Laplacian Matrix to perform very efficient GPU operations. However, this model has a limited expressivity as the convolutions are performed by a for of "averaging" of the neighbouring features using Laplacian matrices.

Average Time 0.27200 s. It corresponds to the dense version as implemented in the public code of [MoNet]. TensorFlow uses GPU computation with dense adjacency matrices. The main drawback is the memory requirements which limit the graph size and the number of features.

Average Time 1.67996 s. TensorFlow cannot use the GPU to perform some sparse matrix operations which related to the construction of the sparse geometric weight matrix used in Geometric Convolutions.

The efficiency and speed achieved by sparse matrix multiplications in [gcn], calls for a Sparse GPU implementation of the Geometric Convolutions [MoNet].

[gcn] : Thomas N. Kipf, Max Welling, Semi-Supervised Classification with Graph Convolutional Networks, ICLR 2017 [Monet] : F. Monti*, D. Boscaini*, J. Masci, E. Rodolà, J. Svoboda, M. M. Bronstein, Geometric deep learning on graphs and manifolds using mixture model CNNs, CVPR 2017 (MoNet framework)