Here we approach the two-class classification problem in a direct way:

- We try and find a plane that separates the classes in feature space

If we cannot, we get creative in two ways

We soften what we mean by “separates”

We enrich and enlarge the feature space so that separation is possible

A hyperplane in \(p\) dimensions is a flat affine subspace of dimension \(p − 1\)

In general the equation for a hyperplane has a form:

- \(\beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_pX_p= 0\)

In \(p=2\) dimensions a hyperplane is a line

If \(\beta_0=0\), the hyperplane goes through the origin, otherwise not

The vector \(\beta = (\beta_1, \beta_2, ..., \beta_p)\) is called the normal vector

- It points in a direction orthogonal to the surface of a hyperplane

Hyperplane in 2 Dimensions