Which of the following weights generates the absolute value function, y = [x], for the following neural network:

h₁Relu(W11x), h₂ = ReLu(W12x), and y ReLu(W21h1+ W22h2). where the input is x and output is y? wij are the parameter values.

The best [W11. W12, W21, W22] combination is:

[1,-1,1, 1]

[1,-1,1,-1]

[0.5.0.5, -2, 2]

[1.5,0.5, -2, 2]

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.

Explore how to solve a neural network problem involving ReLU activation functions. Understand how specific weight combinations affect the output, using a detailed step-by-step analysis. Suitable for advanced learners and professionals.

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