You are given 6 training examples for a binary classification problem as follows:

X1 X2 Y
10 0 +
0 -10 +
5 -2 +
5 10 -
0 5 -
5 5 -

Plot these points in a 2-dimensional grid and show the decision boundary resulting from 1-Nearest Neighbor. Hint: To get started, connect each point with the closest point from the opposite class. Then, for each of the 5 line segments that you drew in this way, draw the perpendicular bisector. The decision boundary consists of certain line segments of these bisectors.

To simplify the discussion, we will only consider two-class classifiers in this section and define a linear classifier as a two-class classifier that decides class membership by comparing a linear combination of the features to a threshold.

Figure 14.8: There are an infinite number of hyperplanes that separate two linearly separable classes.

\includegraphics[width=6cm]{vclassline.eps}

In two dimensions, a linear classifier is a line. Five examples are shown in Figure 14.8 . These lines have the functional form $w_1x_1+w_2x_2=b$. The classification rule of a linear classifier is to assign a document to $c$ if $w_1x_1+w_2x_2>b$ and to $\overline{c}$ if $w_1x_1+w_2x_2\leq b$. Here, $(x_1, x_2)^{T}$ is the two-dimensional vector representation of the document and $(w_1, w_2)^{T}$ is the parameter vector that defines (together with $b$) the decision boundary. An alternative geometric interpretation of a linear classifier is provided in Figure 15.7 (page [*]).

We can generalize this 2D linear classifier to higher dimensions by defining a hyperplane as we did in Equation 140, repeated here as Equation 144:

\begin{displaymath}

\vec{w}^{T}\vec{x} = b

\end{displaymath}(144)

The assignment criterion then is: assign to $c$ if $\vec{w}^{T}\vec{x} > b$ and to $\overline{c}$ if $\vec{w}^{T}\vec{x} \leq b$. We call a hyperplane that we use as a linear classifier a decision hyperplane .

Figure 14.9: Linear classification algorithm.

\begin{figure}\begin{algorithm}{ApplyLinearClassifier}{\vec{w},b,\vec{x}}

score ...

...in{IF}{score>b}

\RETURN{1}

\ELSE

\RETURN{0}

\end{IF}\end{algorithm}

\end{figure}

The corresponding algorithm for linear classification in $M$ dimensions is shown in Figure 14.9 . Linear classification at first seems trivial given the simplicity of this algorithm. However, the difficulty is in training the linear classifier, that is, in determining the parameters $\vec{w}$ and $b$ based on the training set.

Explanation:

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