Outline of the slecture

• Background in Linear Classification Problem
• Support vector machine
• Summary
• References

Background in Linear Classification Problem

In this section, we will introduce the framework and basic idea of linear classification problem.

In a linear classification problem, the feature space can be divided into different regions by hyperplanes. In this lecture, we will take a two-catagory case to illustrate. Given training samples $ \vec{y}_1,\vec{y}_2,...\vec{y}_n \in \mathbb{R}^p $, each $ \vec{y}_i $ is a p-dimensional vector and belongs to either class $ w_1 $ or $ w_2 $. The goal is to find the maximum-margin hyperplane that separate the points in the feature space that belong to class $ w_1 $ from those belong to class$ w_2 $. The discriminate function can be written as

We want to find $ \vec{c}\in\mathbb{R}^{n+1} $ so that a testing data point $ \vec{y}_i $ is labelled

We can apply a trick here to replace all $ \vec{y} $'s in class $ w_2 $ by $ -\vec{y} $, then the task is looking for $ \vec{c} $ so that

Then hyperplane through origin is defined by $ \vec{c}\cdot \vec{y} = 0 $, where \vec{c} is the normal of the plane lying on the positive side of every hyperplane.

You might have already observe the ambiguity of c in the above discussion: if c separates data, $ \lambda \vec{c} $ also separates the data. One solution might be set $ |\vec{c}|=1 $. Another solution is to introduce the concept of "margin" which we denote by b, and ask

In this scenario, $ \frac{b_i}{||c||} $ is the distance from each point to the hyperplane.

However, it is not always possible to find a solution for c. An alternative approach is to find c that minimize a criterion function $ J(\textbf{a}) $that satisfy $ \vec{c}\cdot \vec{y}>0 $.