MATLAB Implementation of Gaussian Kernel Function with Code Descriptions

The Gaussian kernel function is a widely used kernel in machine learning and pattern recognition fields, which constructs non-linear decision boundaries by calculating similarity between samples. In MATLAB implementation, the Gaussian kernel primarily involves two key parameters: kernel width (also known as bandwidth parameter) and distance calculation between input vectors. The implementation typically uses vectorized operations for efficient distance computation.

The core formula of Gaussian kernel function is based on an exponential decay function, whose mathematical expression relies on the Euclidean distance between input vectors. In MATLAB implementation, we first compute the squared distance matrix between vectors using efficient matrix operations like pdist2 or manual vectorization, then apply scaling transformation and exponential operation to these distances. The kernel width parameter controls the smoothness of the function - smaller values create more complex decision boundaries while larger values produce smoother boundaries. The implementation often uses exp(-gamma * squared_distance) where gamma = 1/(2*sigma^2).

Special attention should be paid to parameter selection in practical applications, as inappropriate parameters may lead to overfitting or underfitting. Parameter optimization can be achieved through methods like cross-validation or grid search. When used in algorithms like Support Vector Machines (SVM), the Gaussian kernel can effectively handle non-linearly separable data, though computational complexity increases significantly with sample size due to O(n^2) distance matrix calculations.

The Gaussian kernel function finds extensive applications in various domains including image processing, bioinformatics, and financial modeling. Its MATLAB implementation can be easily integrated into existing classification or regression workflows using built-in functions like fitcsvm for SVM with kernel options, or through custom kernel function implementations.