  Efficient CNN designs like ResNets and DenseNet were proposed to improve
accuracy vs efficiency trade-offs. They essentially increased the connectivity,
allowing efficient information flow across layers. Inspired by these
techniques, we propose to model connections between filters of a CNN using
graphs which are simultaneously sparse and well connected. Sparsity results in
efficiency while well connectedness can preserve the expressive power of the
CNNs. We use a well-studied class of graphs from theoretical computer science
that satisfies these properties known as Expander graphs. Expander graphs are
used to model connections between filters in CNNs to design networks called
X-Nets. We present two guarantees on the connectivity of X-Nets: Each node
influences every node in a layer in logarithmic steps, and the number of paths
between two sets of nodes is proportional to the product of their sizes. We
also propose efficient training and inference algorithms, making it possible to
train deeper and wider X-Nets effectively.
  Expander based models give a 4% improvement in accuracy on MobileNet over
grouped convolutions, a popular technique, which has the same sparsity but
worse connectivity. X-Nets give better performance trade-offs than the original
ResNet and DenseNet-BC architectures. We achieve model sizes comparable to
state-of-the-art pruning techniques using our simple architecture design,
without any pruning. We hope that this work motivates other approaches to
utilize results from graph theory to develop efficient network architectures.
