Math_problem4part2
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Apr 3, 2024
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To construct a hard margin support vector machine (SVM) classifier, we need to first understand the basic concept and then use the given support vectors to define the decision boundary.
### Hard Margin Support Vector Machine
In a hard margin SVM, the goal is to find a hyperplane that separates the classes with the maximum margin and with no misclassifications. Mathematically, given training data with features \(x_i\) and labels \(y_i\), where \(y_i \in \{-1, 1\}\), the objective is to find the hyperplane \(w \cdot x +
b = 0\) such that:
1. \(w \cdot x_i + b \geq 1\) for all \(x_i\) of class 1
2. \(w \cdot x_i + b \leq -1\) for all \(x_i\) of class -1
The margin between the two classes is \(\frac{2}{\|w\|}\), which needs to be maximized.
### Given Support Vectors
Let's say the support vectors are denoted as \(SV_1\), \(SV_2\), and \(SV_3\), where \(SV_1\) belongs to class 1 and \(SV_2\), \(SV_3\) belong to class 2.
### Steps to Construct the Hard Margin Classifier:
1. Compute the hyperplane equation using the support vectors.
2. Determine the normal vector \(w\) and the bias term \(b\) of the hyperplane.
### Solution Steps:
1. **Compute the hyperplane equation:**
Since \(SV_1\) belongs to class 1 and \(SV_2\), \(SV_3\) belong to class 2, the hyperplane should satisfy:
- \(w \cdot SV_1 + b = 1\)
- \(w \cdot SV_2 + b = -1\)
- \(w \cdot SV_3 + b = -1\)
2. **Determine \(w\) and \(b\):**
We can solve the system of equations to find \(w\) and \(b\).
Let's proceed with the solution by assuming the coordinates of support vectors are known.
### Example Solution (Assuming Known Coordinates):
Let's say the coordinates of the support vectors are:
- \(SV_1 = (x_1, y_1)\)
- \(SV_2 = (x_2, y_2)\)
- \(SV_3 = (x_3, y_3)\)
Using these coordinates, we can compute \(w\) and \(b\) to define the hyperplane.
Please provide the coordinates of the support vectors to proceed with the calculations.
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