We present a new convolution layer for deep learning architectures which we
call QuadConv — an approximation to continuous convolution via quadrature. Our
operator is developed explicitly for use on unstructured data, and accomplishes
this by learning a continuous kernel that can be sampled at arbitrary
locations. In the setting of neural compression, we show that a QuadConv-based
autoencoder, resulting in a Quadrature Convolutional Neural Network (QCNN), can
match the performance of standard discrete convolutions on structured uniform
data, as in CNNs, and maintain this accuracy on unstructured data.