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

Function Space and Critical Points of Linear Convolutional Networks