GATE 2025 DA Q.55: solution and answer

GATE 2025 · DAQ.55 · 2 marks · MSQ

Consider a two-class problem in $R^{d}$ with class labels red and green. Let $μ_{r e d}$ and $μ_{g r ee n}$ be the means of the two classes. Given test sample $x \in R^{d}$ , a classifier calculates the squared Euclidean distance (denoted by $∥ \cdot ∥^{2}$ ) between $x$ and the means of the two classes and assigns the class label that the sample $x$ is closest to. That is, the classifier computes $f (x) = ∥ μ_{r e d} - x ∥^{2} - ∥ μ_{g r ee n} - x ∥^{2}$ and assigns the label red to $x$ if $f (x) < 0$ , and green otherwise. Which of the following statements is/are correct?

(A)
The sample $x = 0$ is assigned the label green if $∥ μ_{r e d} ∥ < ∥ μ_{g r ee n} ∥$
(B)
$f$ is a linear function of $x$
(C)
$f (x) = w^{⊤} x + b$ , where $w$ and $b$ are functions of $μ_{r e d}$ and $μ_{g r ee n}$
(D)
$f$ is a quadratic polynomial in $x$

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Answer and solution

ANSWER

The correct options are (B) and (C): $f (x) = w^{⊤} x + b$ with $w = 2 (μ_{g r ee n} - μ_{r e d})$ and $b = ∥ μ_{r e d} ∥^{2} - ∥ μ_{g r ee n} ∥^{2}$ .

Official answer: B, CSource: Official answer key, GATE 2025 DA (IIT Roorkee)

Expand each squared distance: $∥ μ - x ∥^{2} = ∥ μ ∥^{2} - 2 μ^{⊤} x + ∥ x ∥^{2}$ .
Subtract the two expansions. The $∥ x ∥^{2}$ terms cancel, so $f (x) = ∥ μ_{r e d} ∥^{2} - ∥ μ_{g r ee n} ∥^{2} - 2 μ_{r e d}^{⊤} x + 2 μ_{g r ee n}^{⊤} x .$
Hence $f (x) = w^{⊤} x + b$ with $w = 2 (μ_{g r ee n} - μ_{r e d})$ and $b = ∥ μ_{r e d} ∥^{2} - ∥ μ_{g r ee n} ∥^{2}$ . Both $w$ and $b$ depend only on the two class means, so $f$ is linear (affine) in $x$ : (B) and (C) are correct.
(D): no term of degree 2 in $x$ survives the subtraction, so $f$ has degree 1 and is not a quadratic polynomial. (D) is false.
(A): at $x = 0$ , $f (0) = b = ∥ μ_{r e d} ∥^{2} - ∥ μ_{g r ee n} ∥^{2}$ . If $∥ μ_{r e d} ∥ < ∥ μ_{g r ee n} ∥$ then $f (0) < 0$ , so the label is red, not green. (A) is false.
Correct options: B and C.

KEY IDEA

The nearest-mean classifier with squared Euclidean distance has a linear decision rule: the $∥ x ∥^{2}$ terms cancel in the difference, leaving a hyperplane boundary $w^{⊤} x + b = 0$ .

WATCH OUT FOR

Seeing squared distances and concluding $f$ is quadratic in $x$ . The quadratic term $∥ x ∥^{2}$ is identical in both distances and cancels. Another slip is reading $f (0) < 0$ as green, when negative $f$ means red.

From the GATE 2025 DA paper. Every question of this paper · Download the full paper · All 7 LDA, SVM & trees questions, with notes

GATE 2025 DA Q.55: solution and answer

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