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A linear classifier in SVM can be mathematically expressed as [tex]f(x) = \sum_i w_i \Phi_i(x)[/tex], where each [tex]\Phi_i(x)[/tex] is a feature function of the original feature set [tex]x[/tex]. Note that the transformed feature set [tex]\left\{\Phi_i(x)\right\}[/tex] could be infinite-dimensional. The predicted class for a test instance [tex]x[/tex] is determined as follows:

[tex]\[
\hat{y} =
\begin{cases}
+1, & \text{if } f(x) \geq 0 \\
-1, & \text{otherwise}
\end{cases}
\][/tex]

For each binary classification data set described below, state whether it can be perfectly classified by a linear classifier by choosing appropriate feature functions. Restrict the feature functions to polynomial expansions of the original attributes, e.g., [tex]\Phi_1(x) = x_1 x_2^2[/tex] or [tex]\Phi_2(x) = x_2 x_3 x_4[/tex], where [tex]x_1, x_2, x_3[/tex], and [tex]x_4[/tex] are part of the original attributes. If the answer is yes, write the mathematical expression for the linear classifier [tex]f(x)[/tex]. Identify the feature functions [tex]\Phi_i(x)[/tex] as well as the parameters [tex]w[/tex] in your expression.

(a) A data set with 4 continuous-valued features [tex]x_1, x_2, x_3[/tex], and [tex]x_4[/tex]. The class label is +1 if the product of [tex]x_1[/tex] and [tex]x_2[/tex] is greater than or equal to the product of [tex]x_3[/tex] and [tex]x_4[/tex]; otherwise, it is -1. (3 points)



Answer :

GinnyAnswer
To determine whether the given data set can be perfectly classified by a linear classifier using polynomial expansions of the original attributes, let's proceed step by step:

### Problem Description
The data set has 4 continuous-valued features: [tex]\( x_1, x_2, x_3, x_4 \)[/tex]. The classification rule is:
- The class label is [tex]\( +1 \)[/tex] if the product [tex]\( x_1 \cdot x_2 \)[/tex] is greater than or equal to the product [tex]\( x_3 \cdot x_4 \)[/tex].
- Otherwise, the class label is [tex]\( -1 \)[/tex].

We need to select appropriate polynomial feature functions and parameters for the linear classifier [tex]\( f(x) \)[/tex] such that it can perfectly classify the data based on the given rule.

### Feature Functions and Linear Classifier
1. Feature Functions: We choose polynomial feature functions based on the products given in the classification rule:
- Let [tex]\( \Phi_1(x) \)[/tex] be the product [tex]\( x_1 \cdot x_2 \)[/tex].
- Let [tex]\( \Phi_2(x) \)[/tex] be the product [tex]\( x_3 \cdot x_4 \)[/tex].

So, the feature transformation is:
[tex]\[ \Phi_1( x ) = x_1 \cdot x_2 \][/tex]
[tex]\[ \Phi_2( x ) = x_3 \cdot x_4 \][/tex]

2. Linear Classifier: The linear classifier can be expressed as:
[tex]\[ f( x ) = w_1 \Phi_1( x ) + w_2 \Phi_2( x ) \][/tex]

Here, we need the classifier to return [tex]\( +1 \)[/tex] if [tex]\(\Phi_1(x) \geq \Phi_2(x)\)[/tex] and [tex]\(-1\)[/tex] otherwise.

To achieve this, we set the weights as follows:
[tex]\[ w_1 = 1 \quad \text{and} \quad w_2 = -1 \][/tex]

Therefore, the expression for the linear classifier becomes:
[tex]\[ f( x ) = (\Phi_1( x )) - (\Phi_2( x )) = (x_1 \cdot x_2) - (x_3 \cdot x_4) \][/tex]

3. Predicted Class: The predicted class [tex]\(\hat{y}\)[/tex] is determined as follows:
[tex]\[ \hat{y} = \begin{cases} +1 & \text{if } f( x ) \geq 0 \\ -1 & \text{otherwise} \end{cases} \][/tex]

### Example
Consider an example input [tex]\( x = [2, 3, 1, 5] \)[/tex]:
- Compute the feature functions:
[tex]\[ \Phi_1( x ) = 2 \cdot 3 = 6 \][/tex]
[tex]\[ \Phi_2( x ) = 1 \cdot 5 = 5 \][/tex]

- Calculate [tex]\( f( x ) \)[/tex]:
[tex]\[ f( x ) = 6 - 5 = 1 \][/tex]

- Determine the predicted class:
[tex]\[ \hat{y} = \begin{cases} +1 & \text{if } 1 \geq 0 \\ -1 & \text{otherwise} \end{cases} \][/tex]
Therefore, [tex]\( \hat{y} = +1 \)[/tex].

### Conclusion
Yes, the given data set can be perfectly classified by a linear classifier with the following feature functions and weights:
- Feature functions:
[tex]\[ \Phi_1( x ) = x_1 \cdot x_2 \][/tex]
[tex]\[ \Phi_2( x ) = x_3 \cdot x_4 \][/tex]
- Weights:
[tex]\[ w_1 = 1 \quad \text{and} \quad w_2 = -1 \][/tex]
- Linear classifier:
[tex]\[ f( x ) = (x_1 \cdot x_2) - (x_3 \cdot x_4) \][/tex]

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