Construct a feed-forward network with four input nodes and three hidden nodes and three output nodes that has lateral inhibition structure in the output layer.

To construct a feed-forward neural network with the specified characteristics, follow these steps:

1. Network Architecture Design

   1. Input Layer: Start with four input nodes. These nodes will receive the input features of your data.
   2. Hidden Layer: Add a hidden layer with three nodes. This layer will process the inputs through weighted connections and activation functions.
   3. Output Layer: Create an output layer with three nodes. This layer will produce the final output of the network.

2. Lateral Inhibition in Output Layer

   1. Concept of Lateral Inhibition: Lateral inhibition is a process where neurons in the output layer inhibit the activity of neighboring neurons. This can enhance contrast and improve the network's ability to distinguish between different outputs.
   2. Implementation: To implement lateral inhibition, you can introduce inhibitory connections between the output nodes. For example, the activation of one output node can reduce the activation of the others. This can be achieved by adding negative weights between the output nodes.

3. Network Connections and Weights

   1. Input to Hidden Layer: Connect each input node to every hidden node. Assign initial weights to these connections, which can be adjusted during training.
   2. Hidden to Output Layer: Connect each hidden node to every output node. Assign initial weights to these connections as well.
   3. Inhibitory Connections in Output Layer: Add inhibitory connections between the output nodes. For instance, if you have output nodes $O_1$, $O_2$, and $O_3$, you can have connections like $O_1 \rightarrow O_2$, $O_1 \rightarrow O_3$, $O_2 \rightarrow O_1$, etc., with negative weights.

4. Activation Functions

   1. Hidden Layer Activation: Use an activation function like ReLU (Rectified Linear Unit) or sigmoid for the hidden layer nodes to introduce non-linearity.
   2. Output Layer Activation: Depending on the task, use a suitable activation function for the output layer. For classification tasks, softmax is commonly used to convert the outputs into probabilities.

5. Training the Network

   1. Forward Propagation: Pass the input data through the network to compute the outputs.
   2. Backward Propagation: Use a loss function to calculate the error and adjust the weights using an optimization algorithm like gradient descent.
   3. Inhibition Adjustment: During training, ensure that the inhibitory connections are adjusted to effectively implement lateral inhibition.

Final Answer

Construct a feed-forward neural network with four input nodes, three hidden nodes, and three output nodes. Implement lateral inhibition in the output layer by adding inhibitory connections with negative weights between the output nodes. Use appropriate activation functions and train the network using forward and backward propagation.