feat: add Radial Basis Function Neural Network (Implements #12322)

[Monasri29-hub]

Description

Implements #12322

This PR adds a complete implementation of Radial Basis Function Neural Network (RBFNN) to the machine learning algorithms collection.

Implementation Details

Architecture:

• Input layer accepts n-dimensional vectors
• Hidden layer uses Gaussian RBF activation functions
• Output layer performs linear combination of hidden activations

Features:

• __init__(num_centers, gamma): Initialize network parameters
• train(X, y): Train using KMeans clustering and least-squares
• predict(X): Make predictions on new data
• Comprehensive error handling and input validation
• Detailed doctests for all methods

Training Process:

1. Center Selection: Uses KMeans clustering to find optimal RBF centers
2. Activation Computation: Calculates Gaussian RBF activations for all samples
3. Weight Optimization: Solves for output weights using least-squares regression

Mathematical Foundation

The Gaussian RBF is defined as: φ(x) = exp(-γ * ||x - c||²)

Where:

• x: input vector
• c: RBF center
• γ: spread parameter

Output is computed as weighted sum: y = Σ(w_i * φ_i(x))

Testing

All doctests pass successfully: python -m doctest machine_learning/rbf_network.py -v

Test coverage:

• Initialization with valid parameters
• Training with various data shapes
• Prediction after training
• Error handling for invalid inputs

Requirements Met

✅ RBFNN class with num_centers and gamma initialization
✅ train(X, y) method with KMeans and least-squares
✅ predict(X) method for making predictions
✅ Gaussian RBF activation functions
✅ Comprehensive documentation and doctests

Example Usage

import numpy as np from machine_learning.rbf_network import RadialBasisFunctionNetwork

Generate training data X_train = np.random.randn(100, 2) y_train = np.sum(X_train ** 2, axis=1)

Create and train RBFNN rbfnn = RadialBasisFunctionNetwork(num_centers=10, gamma=1.0) rbfnn.train(X_train, y_train)

Make predictions X_test = np.random.randn(20, 2) predictions = rbfnn.predict(X_test)

References

• Wikipedia: https://en.wikipedia.org/wiki/Radial_basis_function_network
• Original paper: Broomhead & Lowe (1988)