To solve this problem, we need to analyze how the neural network computes the output $y = \text{ReLU}(W_{21}h_1 + W_{22}h_2)$ based on the given inputs and weight combinations.

The network structure is as follows:

1. First Layer:

   • $h_1 = \text{ReLU}(W_{11}x)$
   • $h_2 = \text{ReLU}(W_{12}x)$

2. Output Layer:

   • $y = \text{ReLU}(W_{21}h_1 + W_{22}h_2)$

Step-by-Step Analysis:

1. Activation Functions $h_1$ and $h_2$:

   • Since $h_1 = \text{ReLU}(W_{11}x)$, $h_1$ will be $W_{11}x$ if $W_{11}x \geq 0$, and 0 otherwise.
   • Similarly, $h_2 = \text{ReLU}(W_{12}x)$ will be $W_{12}x$ if $W_{12}x \geq 0$, and 0 otherwise.

2. Output $y$:

   • The output $y = \text{ReLU}(W_{21}h_1 + W_{22}h_2)$ depends on the values of $h_1$ and $h_2$.

To generate the absolute value function $y = |x|$, the network needs to correctly sum the contributions from $h_1$ and $h_2$ such that the output is always non-negative and equal to $|x|$.

Test the given combinations:

1. [1, -1, 1, 1]:

   • $h_1 = \text{ReLU}(1 \cdot x) = \text{ReLU}(x) = x$ if $x \geq 0$, 0 otherwise.
   • $h_2 = \text{ReLU}(-1 \cdot x) = \text{ReLU}(-x) = -x$ if $x \leq 0$, 0 otherwise.
   • $y = \text{ReLU}(1 \cdot h_1 + 1 \cdot h_2) = \text{ReLU}(x - x) = 0$. This does not match $|x|$.

2. [1, -1, 1, -1]:

   • $h_1 = \text{ReLU}(1 \cdot x) = x$ if $x \geq 0$, 0 otherwise.
   • $h_2 = \text{ReLU}(-1 \cdot x) = -x$ if $x \leq 0$, 0 otherwise.
   • $y = \text{ReLU}(1 \cdot h_1 - 1 \cdot h_2) = \text{ReLU}(x + x) = |x|$. This matches the function $y = |x|$.

3. [0.5, 0.5, -2, 2]:

   • $h_1 = \text{ReLU}(0.5 \cdot x) = 0.5x$ if $x \geq 0$, 0 otherwise.
   • $h_2 = \text{ReLU}(0.5 \cdot x) = 0.5x$ if $x \geq 0$, 0 otherwise.
   • $y = \text{ReLU}(-2 \cdot 0.5x + 2 \cdot 0.5x) = \text{ReLU}(0) = 0$. This does not match $|x|$.

4. [1.5, 0.5, -2, 2]:

   • $h_1 = \text{ReLU}(1.5 \cdot x) = 1.5x$ if $x \geq 0$, 0 otherwise.
   • $h_2 = \text{ReLU}(0.5 \cdot x) = 0.5x$ if $x \geq 0$, 0 otherwise.
   • $y = \text{ReLU}(-2 \cdot 1.5x + 2 \cdot 0.5x) = \text{ReLU}(-3x + x) = \text{ReLU}(-2x)$, which is 0 when $x > 0$, and non-zero otherwise. This does not match $|x|$.

Conclusion:

The correct combination that generates the absolute value function $y = |x|$ is:

[1, -1, 1, -1]

Would you like further details or have any other questions?

Relative Questions:

1. What is the role of the ReLU activation function in neural networks?
2. How can we design a neural network to approximate non-linear functions?
3. Why does the combination [1, -1, 1, -1] correctly compute the absolute value function?
4. What would happen if we used a different activation function, like a sigmoid, instead of ReLU?
5. How does the choice of weights affect the output of a neural network?

Tip: When designing neural networks, carefully choose weights to shape the output, especially when approximating specific functions.