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],[−1,1],[1,−1]
Class 𝐶2:[1,1]
The decision boundary between the two classes 𝐶1 and 𝐶2 using single perception is given by :
- A.
\(x_1-x_2-0.5=0\) - B.
\(-x_1-x_2-0.5=0\) - C.
\(0.5(x_1+x_2)-1.5=0\) - D.
\(x_1+x_2-0.5=0\)
Attempted by 51 students.
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Correct answer: D
Decision boundary: x1 + x2 - 0.5 = 0
Reasoning: We need a single linear boundary w1 x1 + w2 x2 + b = 0 that places the point (1,1) on one side and the three points (-1,-1), (-1,1), (1,-1) on the other. Choose w1 = 1, w2 = 1 and b = -0.5, which gives the line x1 + x2 - 0.5 = 0.
Evaluate the expression x1 + x2 - 0.5 for each point:
(1,1): 1 + 1 - 0.5 = 1.5 > 0 (this point is on the positive side)
(-1,-1): -1 + -1 - 0.5 = -2.5 < 0
(-1,1): -1 + 1 - 0.5 = -0.5 < 0
(1,-1): 1 + -1 - 0.5 = -0.5 < 0
Since the single linear boundary x1 + x2 - 0.5 = 0 yields positive value for the point from class C2 and negative values for all points from class C1, it correctly separates the two classes.