Recent works in geometric deep learning have introduced neural networks that allow performing inference tasks on three-dimensional geometric data by defining convolution --and sometimes pooling-- operations on triangle meshes. These methods, however, either consider the input mesh as a graph, and do not exploit specific geometric properties of meshes for feature aggregation and downsampling, or are specialized for meshes, but rely on a rigid definition of convolution that does not properly capture the local topology of the mesh. We propose a method that combines the advantages of both types of approaches, while addressing their limitations: we extend a primal-dual framework drawn from the graph-neural-network literature to triangle meshes, and define convolutions on two types of graphs constructed from an input mesh. Our method takes features for both edges and faces of a 3D mesh as input, and dynamically aggregates them using an attention mechanism. At the same time, we introduce a pooling operation with a precise geometric interpretation, that allows handling variations in the mesh connectivity by clustering mesh faces in a task-driven fashion. We provide theoretical insights of our approach using tools from the mesh-simplification literature. In addition, we validate experimentally our method in the tasks of shape classification and shape segmentation, where we obtain comparable or superior performance to the state of the art.

Additional Feedback: Overall, I tend to reject this paper (5 leaning towards 4) for two reasons: 1. that there is very limited novelty, using the model proposed in [20] and the features proposed in [12]. I will be willing to adjust my rating if the authors could convince me that their method is actually critical to the performance increases (this invalidates 2. while making 1. a much lesser concern). Additional comments: - I am not sure if the proposed architecture leverages any properties of a (manifold or not) mesh. It seems that this primal-dual formulation is applicable to any graphs, despite that the fact that some operations might be equivalent to certain mesh operations. How is the method different from graph convolutional networks on arbitrary graphs?

Two referees are very positive about this paper and recommend acceptance, whereas two referees lean towards rejection. All referees agree that primal-dual graph networks have not been previously investigated for mesh processing, but disagree on whether the contribution should be considered significant and/or incremental. The rebuttal attempts to address this concern by highlighting that primal-dual approaches have only been applied on generic graph benchmarks and further emphasizing that the application of such approaches to meshes is not straightforward and only constitutes one part of the proposed approach. R1 and R4 raised concerns about the experimental validation. The rebuttal only partially addressed their concerns, but after discussion, R4 is convinced that the experimental validation is compelling enough.

Recent works in geometric deep learning have introduced neural networks that allow performing inference tasks on three-dimensional geometric data by defining convolution --and sometimes pooling-- operations on triangle meshes. These methods, however, either consider the input mesh as a graph, and do not exploit specific geometric properties of meshes for feature aggregation and downsampling, or are specialized for meshes, but rely on a rigid definition of convolution that does not properly capture the local topology of the mesh. We propose a method that combines the advantages of both types of approaches, while addressing their limitations: we extend a primal-dual framework drawn from the graph-neural-network literature to triangle meshes, and define convolutions on two types of graphs constructed from an input mesh. Our method takes features for both edges and faces of a 3D mesh as input, and dynamically aggregates them using an attention mechanism. At the same time, we introduce a pooling operation with a precise geometric interpretation, that allows handling variations in the mesh connectivity by clustering mesh faces in a task-driven fashion.