Description of Common Kernel Functions with 3D Visualization

This algorithm primarily describes current common kernel functions and displays their three-dimensional graphical representations. Before explaining the basic concepts of kernel functions, let us first understand what Support Vector Machines (SVM) are. SVM is a widely used classification algorithm whose main principle involves finding an optimal hyperplane in high-dimensional space to achieve classification. The kernel function serves as a crucial component in SVM, responsible for mapping data from low-dimensional space to high-dimensional space, thereby enabling more effective classification. From an implementation perspective, kernel functions operate by computing inner products in transformed feature spaces without explicitly performing the high-dimensional mapping. Common kernel types include: - Linear Kernel: K(x,y) = xᵀy, implemented as a simple dot product - Polynomial Kernel: K(x,y) = (γxᵀy + r)^d, where parameters γ, r, and d control the mapping complexity - Gaussian (RBF) Kernel: K(x,y) = exp(-γ||x-y||²), utilizing Euclidean distance with bandwidth parameter γ In this article, we will introduce these common kernel function types and intuitively demonstrate their characteristics and roles through 3D visualization techniques. The visualizations will show how different kernels transform the feature space and affect the decision boundaries in SVM classification. We hope this article will help readers better understand the concepts and applications of kernel functions in machine learning implementations.