Title: Parameterized Pseudo-Differential Operators for Graph Convolutional Neural Networks. Abstract: We present a novel graph convolutional layer that is fast, conceptually simple, and provides high accuracy with reduced overfitting. Based on pseudo-differential operators, our layer operates on graphs with relative position information available for each pair of connected nodes. We evaluate our method on a variety of supervised learning tasks, including superpixel image classification using the MNIST, CIFAR10, and CIFAR100 superpixel datasets, node correspondence using the FAUST dataset, and shape classification using the ModelNet10  dataset. The new layer outperforms multiple recent architectures on superpixel image classification tasks using the MNIST and CIFAR100 superpixel datasets and performs comparably with recent results on the CIFAR10 superpixel dataset. We measure test accuracy without bias to the test set by selecting the model with the best training accuracy. The new layer achieves a test error rate of 0.80% on the MNIST superpixel dataset, beating the closest reported rate of 0.95% by a factor of more than 15%. After dropping roughly 70% of the edge connections from the input by performing a Delaunay triangulation, our model still achieves a competitive error rate of 1.04%.