Normalized Augmented Feature Vector

Used in Classification, where we want to separate different classes.

Remember that the linear discriminant function is

$$g(\mathbf{x})={\mathbf{w}}^T\mathbf{x}+w_0={\mathbf{\alpha }}^T\mathbf{y}$$

where

$$\mathbf{y}=\left[\begin{array}{c}1\\ \mathbf{x}\end{array}\right]$$

and

$$\alpha =\left[\begin{array}{c}{w}_0\\ \mathbf{w}\end{array}\right]$$

In here, note that:

• $g(\mathbf{x})>0$ means that the point lies on the 1st class
• $g(\mathbf{x})<0$ means that that point lies on the 2nd class

We then have to normalize the function using

$${\mathbf{y}}_i^{\prime }=\{\begin{array}{ll}{\mathbf{y}}_i,& \text{if yi∈ω1}\\ -{\mathbf{y}}_i& \text{if yi∈ω2}\end{array}$$

then we would have ${\alpha }^T{\mathbf{y}}_i^{\prime }>0$ for $i=1,\dots ,N$ We call ${\mathbf{y}}_i$ as the normalized augmented feature vector.

It’s very hard-coded to binary classification problems.