Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks

In this work, we theoretically analyze sub-networks and their bias through the lens of algebraic geometry. We consider fully-connected networks with polynomial activation functions, and focus on the geometry of the function space they parametrize, often referred to as neuroman-ifold. First, we compute the dimension of the subspace of the neuromanifold parametrized by subnetworks. Second, we show that this subspace is singular. Third, we argue that such singularities often correspond to critical points of the training dynamics. Lastly, we discuss convolutional networks, for which subnet-works and singularities are similarly related, but the bias does not arise.Figure 1: Subnetworks define singular points (orange) of the neuromanifold.