On Transversality of Bent Hyperplane Arrangements and the Topological Expressiveness of ReLU Neural Networks

Let F : R-n -> R be a feed forward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piece wise affine-linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions, F-1((-infinity, t)) and F -1((t, infinity)), for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a generic, transversal ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially oriented linear 1-complex in the domain of F and identify properties of this complex that yield an obstruction to the existence of bounded connected components of a decision region. We use this obstruction to prove that a decision region of a generic, transversal ReLU network F : R-n -> R with a single hidden layer of dimension n+ 1 can have no more than one bounded connected component.