GAUGE EQUIVARIANT MESH CNNS ANISOTROPIC CONVOLUTIONS ON GEOMETRIC GRAPHS

Abstract

A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.

1. INTRODUCTION

Convolutional neural networks (CNNs) have been established as the default method for many machine learning tasks like speech recognition or planar and volumetric image classification and segmentation. Most CNNs are restricted to flat or spherical geometries, where convolutions are easily defined and optimized implementations are available. The empirical success of CNNs on such spaces has generated interest to generalize convolutions to more general spaces like graphs or Riemannian manifolds, creating a field now known as geometric deep learning (Bronstein et al., 2017) . A case of specific interest is convolution on meshes, the discrete analog of 2-dimensional embedded Riemannian manifolds. Mesh CNNs can be applied to tasks such as detecting shapes, registering different poses of the same shape and shape segmentation. If we forget the positions of vertices, and which vertices form faces, a mesh M can be represented by a graph G. This allows for the application of graph convolutional networks (GCNs) to processing signals on meshes. * Equal Contribution † Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc. The distinct geometry of the neighbourhoods is reflected in the different angles θpq i of incident edges from neighbours qi. Graph convolutional networks apply isotropic kernels and can therefore not distinguish both neighbourhoods. Gauge Equivariant Mesh CNNs apply anisotropic kernels and are therefore sensitive to orientations. The arbitrariness of reference orientations, determined by a choice of neighbour q0, is accounted for by the gauge equivariance of the model. 1



Figure 1: Two local neighbourhoods around vertices p and their representations in the tangent planes TpM .

