Expressiveness of Neural Networks Having Width Equal or Below the Input Dimension

The understanding about the minimum width of deep neural networks needed to ensure universal approximation for different activation functions has progressively been extended (Park et al., 2020). In particular, with respect to approximation on general compact sets in the input space, a network width less than or equal to the input dimension excludes universal approximation. In this work, we focus on network functions of width less than or equal to the latter critical bound. We prove that in this regime, the exact fit of partially constant functions on disjoint compact sets is still possible for ReLU network functions under some conditions on the mutual location of these components. We show that with cosine as activation function, a three layer network of width one is sufficient to approximate any function on arbitrary finite sets. Conversely, we prove a maximum principle from which we conclude that for all continuous and monotonic activation functions, universal approximation of arbitrary continuous functions is impossible on sets that coincide with the boundary of an open set plus an inner point.