Question

Could you please provide the specific details and mathematical expression for the given 2 hidden layer ReLU network with inputs \(x \in \mathbb{R}\), 1-dimensional outputs, and 2 neurons per hidden layer?

Answer 1

A 2 hidden layer ReLU network with 2 neurons per hidden layer, taking 1-dimensional inputs
`\(x \in \mathbb{R}\)` and producing 1-dimensional outputs, can be represented mathematically as follows:

`\[ f(x) = W_2 \cdot \text{ReLU}(W_1 \cdot x + b_1) + b_2 \]`

Step-by-step explanation:

In this neural network architecture, each hidden layer has 2 neurons, and the Rectified Linear Unit (ReLU) activation function is used. Let
`\( W_1 \)`be the weight matrix connecting the input layer to the first hidden layer,
`\( b_1 \)`be the bias vector for the first hidden layer,
`\( W_2 \)`be the weight matrix connecting the first hidden layer to the second hidden layer, and
`\( b_2 \)`be the bias vector for the second hidden layer.

The mathematical expression represents the forward pass of the network. The input
`\( x \)`is multiplied by the weight matrix
`\( W_1 \), the bias \( b_1 \)` is added, and then the ReLU activation function is applied element-wise. This result is then multiplied by
`\( W_2 \), and \( b_2 \)` is added to obtain the final output of the network. The use of ReLU activation allows for non-linear transformations, enhancing the network's capacity to learn complex patterns in the data.