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GATE DA 2024 Q17: Possible support vectors of a hard-margin SVM

Question

Consider the dataset with six datapoints: {(x1,y1),(x2,y2),…,(x6,y6)}\{(\mathbf{x}_1, y_1), (\mathbf{x}_2, y_2), \ldots, (\mathbf{x}_6, y_6)\}, where x1=[10]\mathbf{x}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, x2=[01]\mathbf{x}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, x3=[0−1]\mathbf{x}_3 = \begin{bmatrix} 0 \\ -1 \end{bmatrix}, x4=[−10]\mathbf{x}_4 = \begin{bmatrix} -1 \\ 0 \end{bmatrix}, x5=[22]\mathbf{x}_5 = \begin{bmatrix} 2 \\ 2 \end{bmatrix}, x6=[−2−2]\mathbf{x}_6 = \begin{bmatrix} -2 \\ -2 \end{bmatrix} and the labels are given by y1=y2=y5=1y_1 = y_2 = y_5 = 1, and y3=y4=y6=−1y_3 = y_4 = y_6 = -1. A hard margin linear support vector machine is trained on the above dataset.

Which ONE of the following sets is a possible set of support vectors?

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Answer: D (official GATE 2024 final answer key)

Concept used: Support vectors are the training points that lie on the margin of the maximum-margin separator.

First step: The separator is the line x1+x2=0x_1 + x_2 = 0; the closest points on each side are x1,x2\mathbf{x}_1, \mathbf{x}_2 (positive) and x3,x4\mathbf{x}_3, \mathbf{x}_4 (negative).

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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