Backpropagation Algorithm for Three-Class Classification

The Backpropagation Algorithm for three-class classification is a neural network training method based on backpropagation, suitable for solving multi-class problems. When implemented in MATLAB, it typically involves key steps such as network initialization, forward propagation, error computation, and backward weight updates.

Network Architecture Design The output layer for three-class problems usually employs three neurons, each corresponding to one class. The number of hidden layer nodes can be adjusted based on data complexity. Common activation functions include Sigmoid or ReLU, while the output layer typically uses the Softmax function to ensure probability distribution. In MATLAB implementation, the network structure can be defined using functions like feedforwardnet or through custom layer initialization with specified activation functions.

Forward Propagation Input data is computed layer by layer through the hidden layers, ultimately producing probability values for three classes at the output layer. The output results must match One-Hot encoded labels (e.g., [1,0,0] represents the first class). MATLAB's matrix operations efficiently handle these computations through vectorized implementations, where each layer's output is calculated using matrix multiplication followed by activation function application.

Error Computation and Backpropagation Cross-entropy loss function is used to measure the difference between predictions and true labels. Through the chain rule of differentiation, errors are propagated backward from the output layer to hidden layers, adjusting weights and biases layer by layer. MATLAB's automatic differentiation capabilities or manual gradient computation using matrix operations can efficiently implement these calculations. The backward pass involves computing derivatives of the loss with respect to each parameter and propagating these gradients through the network.

Weight Update Gradient descent or optimization algorithms (such as Adam) are employed to update parameters. Learning rate and iteration count require tuning to avoid overfitting or underfitting. In MATLAB, this can be implemented using optimization functions like trainlm or custom training loops with update rules that adjust parameters based on computed gradients. The update process typically involves subtracting the learning rate multiplied by the gradient from the current weights.

Implementation Key Points Data normalization is necessary to accelerate convergence. Adding regularization terms (such as L2 penalty) improves generalization capability. Classification performance for three classes is evaluated using confusion matrices or accuracy metrics. In MATLAB, these can be implemented using functions like normalize for data preprocessing, adding regularization terms to the loss function, and using confusionmat for performance evaluation.

This algorithm can be encapsulated as modular functions in MATLAB, facilitating extension to higher-dimensional classification tasks. The implementation typically involves creating separate functions for forward propagation, loss computation, backward propagation, and parameter updates, allowing for easy maintenance and scalability.