Consider the two class classification task that consists of the following…
2015
Consider the two class classification task that consists of the following points :
Class 𝐶1: [1 1.5][1 -1.5]Class 𝐶2: [-2 2.5][-2 -2.5]The decision boundary between the two classes using single perceptron is given by :
- A.
𝑥1 + 𝑥2 + 1.5 = 0
- B.
𝑥1 + 𝑥2 − 1.5 = 0
- C.
𝑥1 + 1.5 = 0
- D.
𝑥1 − 1.5 = 0
Attempted by 45 students.
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Correct answer: C
We want a linear decision boundary of the form w·x + b = 0.
Observe the given points: both class 1 points have x1 = 1 and x2 = ±1.5 (symmetric), and both class 2 points have x1 = -2 and x2 = ±2.5 (symmetric). Because x2 values are symmetric within each class, the separating boundary does not need an x2 term and will be vertical (depend only on x1).
Class 1 mean = (1, 0).
Class 2 mean = (-2, 0).
Midpoint between means = ((1 + (-2))/2, 0) = (-0.5, 0).
The perpendicular bisector of the line joining the class means is the natural linear separator. It is vertical and passes through x1 = -0.5, so the decision boundary equation is x1 + 0.5 = 0.
Check: for a class 1 point (1, 1.5), x1 + 0.5 = 1.5 > 0, and for a class 2 point (-2, 2.5), x1 + 0.5 = -1.5 < 0, so the line separates the classes correctly.
Therefore the correct linear decision boundary is x1 + 0.5 = 0.