  # Hyperplane

## What is a Hyperplane?

In mathematics, a hyperplane H is a linear subspace of a vector space V such that the basis of H has cardinality one less than the cardinality of the basis for V.  In other words, if V is an n-dimensional vector space than H is an (n-1)-dimensional subspace.  Examples of hyperplanes in 2 dimensions are any straight line through the origin. In 3 dimensions, any plane containing the origin.  In higher dimensions, it is useful to think of a hyperplane as member of an affine family of (n-1)-dimensional subspaces (affine spaces look and behavior very similar to linear spaces but they are not required to contain the origin), such that the entire space is partitioned into these affine subspaces. This family will be stacked along the unique vector (up to sign) that is perpendicular to the original hyperplane.  This "visualization" allows one to easily understand that a hyperplane always divides the parent vector space into two regions.

In machine learning, it may be useful to employ techniques such as support vector machines to learn hyperplanes to separates the data space for classification. The most common example of hyperplanes in practice is with support vector machines. In this case, learning a hyperplane amounts to learning a linear (often after transforming the space using a nonlinear kernel to lend a linear analysis) subspace that divides the data set into two regions for binary classification.  If the dimensionality of the data set is greater than 2, this may be performed multiple times to achieve a multi-way classification.

### Practical Uses of Hyperplanes with Support Vector Machines

• Natural Language Processing – Support vector machines are useful in both text and hypertext classification, since they reduce the amount of labeled training instances needed. Both for inductive and transductive techniques.
• Image Classification – Using hyperplanes increases image recognition and segmentation accuracy.
• Biosciences

– This technique has proven useful to rapidly classify proteins in unknown substances with high accuracy, which drastically cuts down the research time when studying new compounds for potential health uses.