Hyperplanes and Support Vectors
Support Vector Machines (SVM) are powerful classification algorithms that use the concept of hyperplanes and support vectors to effectively classify data points in high-dimensional spaces. In this section, we will delve into the definitions, significance, and mathematical formulations of hyperplanes and support vectors.
What is a Hyperplane?
A hyperplane is a flat affine subspace of one dimension less than its ambient space. In the context of SVM, it is used to separate different classes of data points. For example, in a 2D space, a hyperplane is simply a line, while in 3D, it is a plane. The general equation of a hyperplane can be expressed as:$$ extbf{w}^T extbf{x} + b = 0 $$
Where: - w is the weight vector that is perpendicular to the hyperplane. - x is the input feature vector. - b is the bias term.
Visualizing Hyperplanes
To visualize, consider a 2D example:
In this image, the line (hyperplane) separates two classes (red and blue) in the feature space. The goal of SVM is to find the optimal hyperplane that maximizes the margin between the classes.