Support Vector Machines

Sanjiv R. Das

What is a SVM?

The goal of the SVM is to map a set of entities with inputs $X=\{x_1,x_2,\ldots,x_n\}$ of dimension $n$, i.e., $X \in R^n$, into a set of categories $Y=\{y_1,y_2,\ldots,y_m\}$ of dimension $m$, such that the $n$-dimensional $X$-space is divided using hyperplanes, which result in the maximal separation between classes $Y$. A hyperplane is the set of points ${\bf x}$ satisfying the equation

$$ {\bf w} \cdot {\bf x} = b $$

where $b$ is a scalar constant, and ${\bf w} \in R^n$ is the normal vector to the hyperplane, i.e., the vector at right angles to the plane. The distance between this hyperplane and ${\bf w} \cdot {\bf x} = 0$ is given by $b/||{\bf w}||$, where $||{\bf w}||$ is the norm of vector ${\bf w}$.

$H_3$ is the best separating hyperplane.

Hyperplane Geometry


$$ \min_{b_1,b_2,{\bf w},\{\eta_i\}} \frac{1}{2} ||{\bf w}||^2 + C_1 \sum_{i=1}^n \eta_i + C_2 \sum_{i=1}^n \eta_i $$

NCAA Dataset