  Recently, there has been a growing focus on determining the minimum width
requirements for achieving the universal approximation property in deep, narrow
Multi-Layer Perceptrons (MLPs). Among these challenges, one particularly
challenging task is approximating a continuous function under the uniform norm,
as indicated by the significant disparity between its lower and upper bounds.
To address this problem, we propose a framework that simplifies finding the
minimum width for deep, narrow MLPs into determining a purely geometrical
function denoted as $w(d_x, d_y)$. This function relies solely on the input and
output dimensions, represented as $d_x$ and $d_y$, respectively. Two key steps
support this framework. First, we demonstrate that deep, narrow MLPs, when
provided with a small additional width, can approximate a $C^2$-diffeomorphism.
Subsequently, using this result, we prove that $w(d_x, d_y)$ equates to the
optimal minimum width required for deep, narrow MLPs to achieve universality.
By employing the aforementioned framework and the Whitney embedding theorem, we
provide an upper bound for the minimum width, given by
$\operatorname{max}(2d_x+1, d_y) + \alpha(\sigma)$, where $0 \leq
\alpha(\sigma) \leq 2$ represents a constant depending on the activation
function. Furthermore, we provide a lower bound of $4$ for the minimum width in
cases where the input and output dimensions are both equal to two.
