Explore the geometry of linear convolutional networks (LCNs) in this 48-minute lecture by Kathlén Kohn at the Hausdorff Center for Mathematics. Delve into the function space of LCNs, identifying it with polynomials that have specific factorizations, and examine how network architecture impacts the geometry of this space. Learn about the characterization of one-dimensional convolutions with stride one and arbitrary filter sizes, including a full description of the function space boundary. Investigate the optimization of objective functions over LCNs, understanding the relationships between critical points in function and parameter spaces, and discover the existence of spurious critical points. Gain insights into upper bounds on critical points in function space using Euclidean distance degrees and explore dynamical invariants for gradient descent. The lecture covers topics such as convolutional matrices, cubic polynomials, optimization of loss functions, and experimental results, providing a comprehensive overview of the geometry of LCNs and their critical points. Read more