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

Let F: Rⁿ → R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise 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((-∞, t)) and F⁻¹((t, ∞)), 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: Rⁿ → R with a single hidden layer of dimension $n+1$ can have no more than one bounded connected component.