Understanding Linear Convolutional Neural Networks via Sparse Factorizations of Real Polynomials

This talk will explain that Convolutional Neural Networks without activation parametrize semialgebraic sets of real homogeneous polynomials that admit a certain sparse factorization. We will investigate how the geometry of these semialgebraic sets (e.g., their singularities and relative boundary) changes with the network architecture. Moreover, we will explore how these geometric properties affect the optimization of a loss function for given training data. This talk is based on joint work with Kathlén Kohn, Guido Montúfar, and Matthew Trager.