Radial Basis Function Kernel
The Radial Basis Function (RBF) kernel, also known as the Gaussian kernel, is one of the most widely used kernel functions. It operates by measuring the similarity between data points based on their Euclidean distance in the input space. Mathematically, the RBF kernel between two data points, [Tex]\mathbf{x}[/Tex] and [Tex]\mathbf{x’}[/Tex], is defined as:
[Tex]K(\mathbf{x}, \mathbf{x’}) = \exp\left(-\frac{|\mathbf{x} – \mathbf{x’}|^2}{2\sigma^2}\right)[/Tex]
where,
- [Tex] |\mathbf{x} – \mathbf{x’}|^2[/Tex] represents the squared Euclidean distance between the two data points.
- [Tex]\sigma[/Tex] is a parameter known as the bandwidth or width of the kernel, controlling the smoothness of the decision boundary.
If we expand the above exponential expression, It will go upto infinite power of x and x’, as expansion of [Tex]e^x[/Tex] contains infinite terms upto infinite power of x hence it involves terms upto infinite powers in infinite dimension.
Radial Basis Function Kernel – Machine Learning
Kernels play a fundamental role in transforming data into higher-dimensional spaces, enabling algorithms to learn complex patterns and relationships. Among the diverse kernel functions, the Radial Basis Function (RBF) kernel stands out as a versatile and powerful tool. In this article, we delve into the intricacies of the RBF kernel, exploring its mathematical formulation, intuitive understanding, practical applications, and its significance in various machine learning algorithms.
Table of Content
- What is Kernel Function?
- Radial Basis Function Kernel
- Transforming Linear Algorithms into Infinite-dimensional Nonlinear Classifiers and Regressors
- Why Radial Basis Kernel Is much powerful?
- Some Complex Dataset Fitted Using RBF Kernel easily:
- Radial Basis Function Neural Network for XOR Classification
- Practical Applications of Radial Basis Function Kernel