Hyperplanes in Higher-Dimensional Spaces: A Practical Explanation with Example and implementation in SVM with 2-D and 3-Dimensional space

In machine learning, classification tasks often require identifying boundaries between classes. One of the most effective ways for achieving this is the hyperplane, especially when used in algorithms like Support Vector Machines (SVM). This post will unpack what a hyperplane is, how it functions mathematically with example, how it defines boundaries between classes, why maximizing the margin around it is key to building robust models, and its implementation in SVM with 2-D and 3-D spaces.

What is a Hyperplane?

A hyperplane is a flat, decision boundary that divides a data space into regions, each associated with a class label. In simple terms, it’s like a line in a two-dimensional space that separates data points of different classes. The concept of a hyperplane extends to higher dimensions: like in three dimensions — it’s a flat plane, and in higher than 3 dimensions — it’s a multi-dimensional space divider.

Image generated in Chatgpt, iluustrates the hyperplane in 3-Dimensioal space

In classification tasks, the hyperplane’s primary role is to distinguish one class from another by creating a separation boundary. Data points lying on one side of the hyperplane are classified as one class, while those on the other side belong to the other class.

Mathematical Definition of a Hyperplane

In an n-dimensional space, a hyperplane is defined by the equation:

Where:

• W is the weight vector (normal to the hyperplane). If its 2-D then weights will W = [w1, w2], and if its 3-D then W = [w1, w2, w3] and for n-Dimesnion then W = [w1, w2, w3, w4, …wn]
• X represents the feature vector of a data point. Similarly for feature vector with n-Dimensions X = [x1, x2, x3, x4, …xn]
• b is the bias term (offset), determining the hyperplane’s position.

In a 2D space, a hyperplane is represented by a line; in a 3D space, it becomes a plane. However, in higher-dimensional spaces (e.g., 4D, 5D, or even 100D), a hyperplane is an (n−1)-dimensional “surface” that separates the data into two classes. Although it is challenging to visualize…