A hyperplane is a flat affine subspace with dimension one less than its ambient space. In d-dimensional feature space, the equation θᵀx + θ₀ = 0 defines a hyperplane determined by normal vector θ and offset θ₀. section 5.2
For a linear classifier, this hyperplane is the decision boundary. The sign of θᵀx + θ₀ assigns a point to one of two labels: points on the normal-vector side receive +1, while points on the opposite side receive −1. section 5.2
Perceptron decision boundary
A perceptron uses θᵀx + θ₀ = 0 to separate two classes. In two dimensions, the boundary appears as a line; in higher-dimensional feature spaces, it remains a codimension-one separating surface. Applying the classifier to iris data produces predictions that can be analyzed with a confusion matrix. section 5.2
Useful geometric tasks include:
- Determine which side of the hyperplane contains a point by evaluating the sign of θᵀx + θ₀. section 5.2
- Compute the Euclidean distance from a point y to the hyperplane θᵀx + θ₀ = 0. section 5.7
