  We propose a generalization of convolutional neural networks (CNNs) to
irregular domains, through the use of a translation operator on a graph
structure. In regular settings such as images, convolutional layers are
designed by translating a convolutional kernel over all pixels, thus enforcing
translation equivariance. In the case of general graphs however, translation is
not a well-defined operation, which makes shifting a convolutional kernel not
straightforward. In this article, we introduce a methodology to allow the
design of convolutional layers that are adapted to signals evolving on
irregular topologies, even in the absence of a natural translation. Using the
designed layers, we build a CNN that we train using the initial set of signals.
Contrary to other approaches that aim at extending CNNs to irregular domains,
we incorporate the classical settings of CNNs for 2D signals as a particular
case of our approach. Designing convolutional layers in the vertex domain
directly implies weight sharing, which in other approaches is generally
estimated a posteriori using heuristics.
