  Stability is a key aspect of data analysis. In many applications, the natural
notion of stability is geometric, as illustrated for example in computer
vision. Scattering transforms construct deep convolutional representations
which are certified stable to input deformations. This stability to
deformations can be interpreted as stability with respect to changes in the
metric structure of the domain. In this work, we show that scattering
transforms can be generalized to non-Euclidean domains using diffusion
wavelets, while preserving a notion of stability with respect to metric changes
in the domain, measured with diffusion maps. The resulting representation is
stable to metric perturbations of the domain while being able to capture
"high-frequency" information, akin to the Euclidean Scattering.
