convolution amounts to filtering the Laplacian eigenvalues, ${p(\Delta) = \Phi p(\Lambda) \Phi^\top}$. Equivalently: $$(f * g) = p(\Delta) f = \sum_{\ell = 0}^r \theta_\ell \Delta^\ell f,$$ While a necessary evil in general graphs, spectral filters on meshes are rather weak given that they are locally rotationally-invariant. On the other hand, spiral convolutional filters leverage the fact that on a mesh one can canonically order the neighbours. Thus, they are anisotropic by construction and as will be shown in the experimental section \[evaluation\] they are expressive by using just one-hop neighbourhoods, contrary to the large receptive fields used in [@coma]. In Fig \[isotropy\] we visualise the impulse response (centred on a vertex on the forehead) of a selected laplacian polynomial filter from the architecture of [@coma] (left) and from a spiral convolutional filter with $h=1$ (right).

Finally, the equivalence of spiral convolutions to image convolutions allows the use of long-studied practices in the computer vision community. For example, small patches can be used, leading to few parameters and fast computation. Furthermore, dilated convolutions [@yu2015multi] can also be adapted to the spiral operator by simply sub-sampling the spiral. Finally, we argue here that our operator could be applied to other domains, such 