using techniques such as compressed sparse row (CSR) for nonlinear quantization. The number of bits needed to store the codebook can also be ignored because it has lower order of complexity.

We consider any function $f$ in the Sobolev space: $f\in\mathscr{W}^{n,\infty}([0,1]^d)$ and $$\max_{\textbf{n}:|\textbf{n}|\leq n}\operatorname*{ess\,sup}_{x\in[0,1]^d}|D^{\textbf{n}}f(\textbf{x})|\leq1.$$ The space $\mathscr{W}^{n,\infty}$ consists of all locally integrable function $f:\Omega\to\mathbb{R}$ such that $D^{\textbf{n}}f\in L^{\infty}(\Omega)$, where $|\textbf{n}|\leq n$ and $\Omega$ is an open set in $\mathbb{R}^d$. We denote this function space as $\mathcal{F}_{d,n}$ in this paper.

Note that we only assume weak derivatives up to order $n$ exist where $n$ can be as small as 1 where the function is non-differentiable. We also only assume the Lipschitz constant to be no greater than 1 for the simplicity of the derivation. When the Lipschitz constant is bigger than 1, as long as it is bounded, the whole flow of the proof remains the same though the bound expression will scale accordingly.

When constructing the network to approximate any target function $f$, we consider two scenarios for deriving the bounds. The first scenario is called function-dependent structure, where the constructed network topology and their associated weights are all affected by the choice of the target function. In contrast, the 