Hyperplane

TL;DR

• A hyperplane is a flat subspace whose dimension is one less than its ambient space and that divides that space into distinct regions.
• Examples: a line in a two-dimensional space, a plane in a three-dimensional space, or a four-dimensional flat surface in a five-dimensional space.
• Common practical use: separating data into classes in machine learning classification problems.

Definition

A hyperplane is a subspace of one dimension less than the space it is embedded in. In other words, a hyperplane is a flat surface that divides a higher-dimensional space into two or more distinct regions.

Explanation

A hyperplane is a flat geometric object that partitions its ambient space into separate regions. In each ambient space, the hyperplane’s dimension is exactly one less than that of the space containing it, and it forms a boundary so that points lie on one side or the other (or on the hyperplane itself). This geometric property makes hyperplanes useful for delineating regions within spaces of any dimension.

Examples

Two-dimensional space

In a two-dimensional space, a line can be thought of as a hyperplane as it divides the space into two distinct regions - one on each side of the line.

Three-dimensional space

In a three-dimensional space, a plane can be considered a hyperplane as it divides the space into two distinct regions - one on each side of the plane.

Higher-dimensional space

In a space with five dimensions, a hyperplane would be a four-dimensional flat surface that divides the space into two or more distinct regions.

Machine learning classification

In machine learning algorithms used for classification, a hyperplane is used to separate data points belonging to different classes by creating a boundary between them. For instance, in a classification problem with two classes, a hyperplane is used to separate the data points belonging to each class by creating a boundary between them. The data points on one side of the hyperplane are classified as belonging to one class, while the data points on the other side are classified as belonging to the other class.

Use cases

• Mathematical and computational applications, such as in machine learning algorithms for classification, where hyperplanes serve as decision boundaries separating categories of data points.

Related terms

• Subspace
• Line
• Plane
• Classification
• Machine learning algorithms
• Flat surface