  Convolutional neural networks and their ilk have been very successful for
many learning tasks involving images. These methods assume that the input is a
scalar image representing the intensity in each pixel, possibly in multiple
channels for color images. In natural-science domains however, image-like data
sets might have vectors (velocity, say), tensors (polarization, say),
pseudovectors (magnetic field, say), or other geometric objects in each pixel.
Treating the components of these objects as independent channels in a CNN
neglects their structure entirely. Our formulation -- the GeometricImageNet --
combines a geometric generalization of convolution with outer products, tensor
index contractions, and tensor index permutations to construct geometric-image
functions of geometric images that use and benefit from the tensor structure.
The framework permits, with a very simple adjustment, restriction to function
spaces that are exactly equivariant to translations, discrete rotations, and
reflections. We use representation theory to quantify the dimension of the
space of equivariant polynomial functions on 2-dimensional vector images. We
give partial results on the expressivity of GeometricImageNet on small images.
In numerical experiments, we find that GeometricImageNet has good
generalization for a small simulated physics system, even when trained with a
small training set. We expect this tool will be valuable for scientific and
engineering machine learning, for example in cosmology or ocean dynamics.
