to $2^{-t}$. Since for any weight in $W_c$ we need to concatenate no more than $\lambda\left(t^{\frac{1}{\lambda-1}}-1\right)$ weights in a straight line, the sub-network has no greater than $\lambda\left(t^{\frac{1}{\lambda-1}}-1\right)+1$ layers, and no greater than $4t\lambda\left(t^{\frac{1}{\lambda-1}}-1\right)+8t+4$ weights.

We now relax the assumption $w\geq0$. When $w<0$, the sub-network can be constructed as $w=|w|$, while we use $-\frac{1}{2}$ instead of $\frac{1}{2}$ in the last layer. To relax the assumption $x\geq0$, we can make a duplication of the sub-network. Let all the weights in the first layer of the sub-network be $\frac{1}{2}$ for one and $-\frac{1}{2}$ for the other. Here we are utilizing the gate property of ReLU. In this way, one sub-network is activated only when $x>0$ and the other is activated only when $x<0$. The sign of the output can be adjusted by flipping the sign of weights in the last layer. Note that the configuration of the sub-network is solely determined by $w$ and works for any input $x$.

The efficiency of the weight approximation is critical to the overall complexity. Compared with the weight selection as $\{2^{-1},2^{{-t\frac{1}{\lambda-1}}},2^{-t{\frac{2}{\lambda-1}}},\dots,2^{-t{\frac{(\lambda-2)}{\lambda-1}}}\}$, our approximation reduces the number of weights by a factor of $t^{\frac{\lambda-2}{\lambda-1}}$.

Approximation of general functions {#general}
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With the help of Proposition \[p2\] 