# Creating a Support Vector Machine in Python

A Support Vector Machine is a discriminative model for classification and regression. SVM's learn linear decision boundaries separating classes, although non-linear boundaries are achievable via the application of non-linear kernel functions to input vectors. The SVM optimization problem seeks to maximize a quantity called margin, intuitively the orthogonal distance between the decision surface and the nearest point in feature space. Suppose we are given training data $\mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^N$ with associated labels $t_1, \ldots, t_N \in \{-1, 1\}$. Then to formulate the model, we make the assumption of a linear hyperplane in feature space defined as $$y(\mathbf{x}) = \mathbf{w}^T\phi(\mathbf{x}) + b = 0$$ We seek to classify each of the $N$ training points correctly, in which case we have $t_ny(\mathbf{x}_n) \geq 1$ for all $n$. The distance between a given point $x$ and the decision boundary is \|y(x)