  We study the geometry of linear networks with one-dimensional convolutional
layers. The function spaces of these networks can be identified with
semi-algebraic families of polynomials admitting sparse factorizations. We
analyze the impact of the network's architecture on the function space's
dimension, boundary, and singular points. We also describe the critical points
of the network's parameterization map. Furthermore, we study the optimization
problem of training a network with the squared error loss. We prove that for
architectures where all strides are larger than one and generic data, the
non-zero critical points of that optimization problem are smooth interior
points of the function space. This property is known to be false for dense
linear networks and linear convolutional networks with stride one.
