| Line 25: | Line 25: | ||
<math> {w_2} </math> if <math> c\cdot\vec{y}<0</math> | <math> {w_2} </math> if <math> c\cdot\vec{y}<0</math> | ||
| − | We can apply a trick here to replace all <math>\vec{y}</math>'s in class <math>w_2</math> by <math>-\vec{y}</math>, then the task is looking for <math>c</math> so that <math>c\cdot \vec{y}>0, \forall \vec{y} \in new | + | We can apply a trick here to replace all <math>\vec{y}</math>'s in class <math>w_2</math> by <math>-\vec{y}</math>, then the task is looking for <math>c</math> so that <math>c\cdot \vec{y}>0, \forall \vec{y} \in </math>new sample space. Then hyperplane through origin is defined by <math>c\cdot \vec{y} = 0</math>, where 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. | ||
Revision as of 12:18, 1 May 2014
'Support Vector Machine and its Applications in Classification Problems
A slecture by Xing Liu
Partially based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
Outline of the slecture
- Linear discriminant functions
- Summary
- References
Linear classification Problem Statement
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
$ g(\vec{y}) = c\cdot\vec{y} $
We want to find $ c\in\mathbb{R}^{n+1} $ so that a testing data point $ \vec{y}_i $ is labelled
$ {w_1} $ if $ c\cdot\vec{y}>0 $
$ {w_2} $ if $ c\cdot\vec{y}<0 $
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 $ c $ so that $ c\cdot \vec{y}>0, \forall \vec{y} \in $new sample space. Then hyperplane through origin is defined by $ c\cdot \vec{y} = 0 $, where 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.
