The Geometry of Linear Convolutional Networks

We discuss linear convolutional neural networks (LCNs) and their critical points. We observe that the function space (i.e., the set of functions represented by LCNs) can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network’s architecture on the geometry of the function space. For instance, for LCNs with one-dimensional convolutions having stride one and arbitrary filter sizes, we provide a full description of the boundary of the function space. We further study the optimization of an objective function over such LCNs: We characterize the relations between critical points in function space and in parameter space and show that there do exist spurious critical points. We compute an upper bound on the number of critical points in function space using Euclidean distance degrees and describe dynamical invariants for gradient descent. This talk is based on joint work with Thomas Merkh, Guido Montúfar, and Matthew Trager.