Improved bounds on the minimal width of universal neural networks via the mod-two double-point obstruction

For integers d,m≥1, we study uniform approximation of continuous maps from the unit d-dimensional cube to Rᵐ by coordinate projections of smooth embeddings. We obtain lower bounds for the geometric approximation width introduced by Hwang. If q is an integer with 0≤ q<d and the q-dimensional real projective space immerses smoothly in Rᵈ with normal bundle whose pullback to the antipodal covering sphere is trivial, the required ambient dimension is at least q+1+min(d,m). In particular, in every input dimension it is at least min(d,m)+ d/2. These estimates imply lower bounds on the minimal hidden width for uniform universal approximation by a certain class of neural networks, including networks with continuous monotone scalar activation functions such as ReLU. Our argument refines the method of Chu--Weinkove by applying the mod-two double-point obstruction to regular antipodal maps in arbitrary codimension.