Linear Classifiers, Discriminant Analysis and Margin-Based Methods Previous Year Questions (PYQs) for GATE DA: 6+ Solved Questions with Step-by-Step Solutions

Solve 6+ Linear Classifiers, Discriminant Analysis and Margin-Based Methods previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Linear Classifiers and Margin Methods

1. Fisher Discriminant and Distance-Based Classifiers

Intuition of class separation, scatter matrices, and optimal projection. Current Topic

2. Fisher Discriminant: Mathematical Optimization

Deriving the weight vector using generalized eigenvalue problems.

3. Linear Separability and Perceptron Updates

Understanding linearly separable data and mistake-driven weight updates.

4. Perceptron Algorithm and Convergence

Formal algorithm steps, learning rate, and convergence theorem.

5. Support Vector Machines: Margins and Vectors

Hard margin, soft margin, and geometric intuition of support vectors.

6. SVM Optimization and Practical Application

Primal and dual formulations, hinge loss, and kernel trick basics.

The Core Idea: Why Fisher Discriminant?

The Goal of Fisher Linear Discriminant (FLD)

When reducing dimensions for classification, maximizing total variance (like PCA) can be disastrous. The direction of maximum variance might be orthogonal to the direction that best separates the classes.

Fisher's insight was to frame dimensionality reduction as a supervised optimization problem. For a binary classification task, we seek a projection vector $w$ that maps $d$ -dimensional data $x$ to a scalar $y = w^{T} x$ , such that:

The projected class means are far apart (maximize between-class scatter).
The projected points within each class are tightly clustered (minimize within-class scatter).

This creates a linear decision boundary in the original space that is optimally tuned for separating the two classes.

Linear Classifiers, Discriminant Analysis and Margin-Based Methods: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Machine Learning MSQ

A.
$w = (44)$ and $b = 1$
B.
The number of support vectors is 3
C.
The margin is $2$
D.
Training accuracy is 98%

Question 2 · Machine Learning MCQ

For any binary classification dataset, let

S_{B} \in R^{d \times d}

and

S_{W} \in R^{d \times d}

be the
between-class and within-class scatter (covariance) matrices, respectively. The
Fisher linear discriminant is defined by

u^{*} \in R^{d}

, that maximizes

J (u) = \frac{u ^{T} S _{B} u}{u ^{T} S _{W} u}

λ = J (u^{*})

S_{W}

is non-singular and

S_{B} \neq = 0

, then

(u^{*}, λ)

must satisfy which ONE
of the following equations?
Note:

R

denotes the set of real numbers.

A.
$S_{W}^{- 1} S_{B} u^{*} = λ u^{*}$
B.
$S_{W} u^{*} = λ S_{B} u^{*}$
C.
$S_{B} S_{W} u^{*} = λ u^{*}$
D.
$u^{* T} u^{*} = λ^{2}$

Question 3 · Machine Learning MSQ

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.
$f (x) = w^{T} x + b$ , where $w$ and $b$ are functions of $μ_{r e d}$ and $μ_{g r ee n}$
D.
$f$ is a quadratic polynomial in $x$

Question 4 · Machine Learning MSQ

Consider the following figures representing datasets consisting of
two-dimensional features with two classes denoted by circles and squares.

Which of the following is/are TRUE?

A.
(i) is linearly separable.
B.
(ii) is linearly separable.
C.
(iii) is linearly separable.
D.
(iv) is linearly separable.

Question 5 · Machine Learning MCQ

Consider designing a linear classifier

y = s i g n (f (x; w, b)), f (x; w, b) = w^{T} x + b

on a dataset

D = {(x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{N}, y_{N})}

x_{i} \in R^{d}

y_{i} \in {+ 1, - 1}

i = 1, 2, ..., N

. Recall that the sign function outputs

+ 1

if the argument is positive, and

- 1

if the argument is non-positive. The parameters

w

and

b

are updated as per the following training algorithm:

w_{n e w} = w_{o l d} + y_{n} x_{n}, b_{n e w} = b_{o l d} + y_{n}

whenever

s i g n (f (x_{n}; w_{o l d}, b_{o l d})) \neq = y_{n}

. In other words, whenever the classifier wrongly predicts a sample

(x_{n}, y_{n})

from the dataset,

w_{o l d}

gets updated to

w_{n e w}

, and likewise

b_{o l d}

gets updated to

b_{n e w}

. Consider the case

(x_{n}, + 1)

f (x_{n}; w_{o l d}, b_{o l d}) < 0

. Then

A.
$f (x_{n}; w_{n e w}, b_{n e w}) > f (x_{n}; w_{o l d}, b_{o l d})$
B.
$f (x_{n}; w_{n e w}, b_{n e w}) < f (x_{n}; w_{o l d}, b_{o l d})$
C.
$f (x_{n}; w_{n e w}, b_{n e w}) = f (x_{n}; w_{o l d}, b_{o l d})$
D.
$y_{n} f (x_{n}; w_{o l d}, b_{o l d}) > 1$

A question from this chapter

Question 1

Question 2

For any binary classification dataset, let S_{B} \in R^{d \times d} and S_{W} \in R^{d \times d} be the between-class and within-class scatter (covariance) matrices, respectively. The Fisher linear discriminant is defined by u^{*} \in R^{d}, that maximizes J (u) = \frac{u ^{T} S _{B} u}{u ^{T} S _{W} u} If λ = J (u^{*}), S_{W} is non-singular and S_{B} \neq = 0, then (u^{*}, λ) must satisfy which ONE of the following equations? Note: R denotes the set of real numbers.

Question 3

Question 4

Consider the following figures representing datasets consisting of two-dimensional features with two classes denoted by circles and squares. (i) (ii) (iii) (iv) 321123321123321123321123 Which of the following is/are TRUE?

Question 5

Consider designing a linear classifier y = s i g n (f (x; w, b)), f (x; w, b) = w^{T} x + b on a dataset D = {(x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{N}, y_{N})}, x_{i} \in R^{d}, y_{i} \in {+ 1, - 1}, i = 1, 2, ..., N . Recall that the sign function outputs + 1 if the argument is positive, and - 1 if the argument is non-positive. The parameters w and b are updated as per the following training algorithm: w_{n e w} = w_{o l d} + y_{n} x_{n}, b_{n e w} = b_{o l d} + y_{n} whenever s i g n (f (x_{n}; w_{o l d}, b_{o l d})) \neq = y_{n} . In other words, whenever the classifier wrongly predicts a sample (x_{n}, y_{n}) from the dataset, w_{o l d} gets updated to w_{n e w}, and likewise b_{o l d} gets updated to b_{n e w} . Consider the case (x_{n}, + 1), f (x_{n}; w_{o l d}, b_{o l d}) < 0 . Then

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