Minimum Width of Deep Narrow Networks for Universal Approximation

Determining the minimum width of fully connected neural networks has become a fundamental problem in recent theoretical studies of deep neural networks. In this paper, we study the lower bounds and upper bounds of the minimum width required for fully connected neural networks in order to have universal approximation capability, which is important in network design and training. We show that $w_{min}\leq\max(2d_x+1, d_y)$ also holds true for networks with ELU, SELU activation functions, and the upper bound of this inequality is attained when $d_y=2d_x$, where $d_x$, $d_y$ denote the input and output dimensions, respectively. Besides, we show that $d_x+1\leq w_{min}\leq d_x+d_y$ for networks with LeakyReLU, ELU, CELU, SELU, Softplus activation functions, by proving that ReLU activation function can be approximated by these activation functions. In addition, in the case that the activation function is injective or can be uniformly approximated by a sequence of injective functions (e.g., ReLU), we present a new proof of the inequality $w_{min}\ge d_y+\mathbf{1}_{d_x<d_y\leq2d_x}$ by constructing a more intuitive example via a new geometric approach based on Poincar\'e-Miranda Theorem.