  Convolutional Neural Networks (CNNs) filter the input data using a series of
spatial convolution operators with compactly supported stencils and point-wise
nonlinearities. Commonly, the convolution operators couple features from all
channels. For wide networks, this leads to immense computational cost in the
training of and prediction with CNNs. In this paper, we present novel ways to
parameterize the convolution more efficiently, aiming to decrease the number of
parameters in CNNs and their computational complexity. We propose new
architectures that use a sparser coupling between the channels and thereby
reduce both the number of trainable weights and the computational cost of the
CNN. Our architectures arise as new types of residual neural network (ResNet)
that can be seen as discretizations of a Partial Differential Equations (PDEs)
and thus have predictable theoretical properties. Our first architecture
involves a convolution operator with a special sparsity structure, and is
applicable to a large class of CNNs. Next, we present an architecture that can
be seen as a discretization of a diffusion reaction PDE, and use it with three
different convolution operators. We outline in our experiments that the
proposed architectures, although considerably reducing the number of trainable
weights, yield comparable accuracy to existing CNNs that are fully coupled in
the channel dimension.
