Algebraic geometry of rational neural networks

Abstract: We study the expressivity of rational neural networks (RationalNets) through the lens of algebraic geometry. We consider rational functions that arise from a given RationalNet to be tuples of fractions of homogeneous polynomials of fixed degrees. For a given architecture, the neuromanifold is the set of all such expressible tuples. For RationalNets with one hidden layer and fixed activation function $1/x$, we characterize the dimension of the neuromanifold and provide defining equations for some architectures. We also propose algorithms that determine whether a given rational function belongs to the neuromanifold. For deep binary RationalNets, i.e., RationalNets all of whose layers except potentially for the last one are binary, we classify when the Zarisky closure of the neuromanifold equals the whole ambient space, and give bounds on its dimensions.