Consider the following dataset with feature vector \mathbf{x} and label y: | \mathbf{x} | y | |---|---| | \begin{bmatrix} 4 & 1 \end{bmatrix}^T | 2 | | \begin{bmatrix} 2 & 8 \end{bmatrix}^T | -14 | | \begin{bmatrix} 1 & 4 \end{bmatrix}^T | -7 | | \begin{bmatrix} 3 & 2 \end{bmatrix}^T | -1 | Based on the above data, answer the given subquestions. Figure from the original question paper

(B) Option figure from the original question paper

(A) Option figure from the original question paper

(C) Option figure from the original question paper

(D) Option figure from the original question paper

May 2025 term · Machine Learning Techniques · BSCS2007

Machine Learning Techniques Quiz 1: 13 July 2025 (May 2025 term)

The IIT Madras BS Machine Learning Techniques (MLT) Quiz 1 paper sat on 13 Jul 2025, in the May 2025 term: 15 questions for 40 marks in 120 minutes. Every question is below with its answer. Take it as a timed mock test to be marked, or read it through first.

Questions: 15
Marks: 40
Duration: 120 min
MCQ: 8
MSQ: 2
Numerical: 5

Take as mock test Practise with answers

Updated 25 Sept 2026

Official paper: IIT M DIPLOMA AN EXAM QDD2 13 July 2025 · No negative marking.

Question 1

+2 marksOne correct option

How does K-means++ algorithm enhance the initialization process as compared to the standard K- means algorithm?

A
It ensures the algorithm converges in a fixed number of iterations.
B
It improves centroid initialization by spreading them out, leading to better separated clusters.
C
It automatically calculates the best value of K.
D
It removes the need to run the algorithm multiple times.

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Correct answer

B
It improves centroid initialization by spreading them out, leading to better separated clusters.

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Question 2

+2 marksOne correct option

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Question 3

+4 marksOne correct option

Let $k : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ be a valid kernel function such that $k(x, y) = \phi(x)^T\phi(y)$ , where $\phi : \mathbb{R}^d \to \mathbb{R}^D$ , where $d < D$ . Define

$k_1(x, y) = \frac{100k(x, y)}{\sqrt{k(x, x)k(y, y)}}.$

Find $\phi_1 : \mathbb{R}^d \to \mathbb{R}^D$ such that $k_1(x, y) = \phi_1(x)^T\phi_1(y)$ .

Note: $||x||_2 = \sqrt{x^Tx}$ .

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Question 3 on its own page

Question 4

+4 marksOne correct option

Kernel PCA with a polynomial kernel is applied on the following dataset with three features:

$X = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}.$

The $k$ -th principal component is given by $w_k = \phi(X)\alpha_k$ , where $\phi$ is the transformation mapping corresponding to the kernel $k(x_i, x_j) = (1 + x_i^Tx_j)^2$ for some $\alpha_k \in \mathbb{R}^n$ . Find the centered kernel matrix $K^c$ .

Hint: $k_{ij}^c = k_{ij} - \theta_i\mathbb{1}_j^T - \mathbb{1}_i\theta_j^T + P\mathbb{1}_i\mathbb{1}_j^T$ , where $\theta_i = \frac{1}{n}\sum_{k=1}^{n} k_{ik}, \forall\, i, P = \frac{1}{n^2}\sum_{i,j} k_{ij}$ .

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Question 5

+3 marksOne correct option

You are presented with a dataset that has latent variables that influences your data. You are asked to use Expectation-Maximization algorithm to best capture the data. How would you define the steps in the EM (Expectation and Maximization) algorithm?

A
E-step: Estimate the latent variables in the dataset, M-step: Maximize the likelihood over the model parameters.
B
E-step: Estimate the number of latent variables in the dataset, M-step: Maximize the likelihood over the model parameters.
C
E-step: Estimate the likelihood over the model parameters, M-step: Maximize the number of latent variables in the dataset.
D
E-step: Estimate the likelihood over the model parameters, M-step: Maximize the number of parameters in the model.

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Correct answer

A
E-step: Estimate the latent variables in the dataset, M-step: Maximize the likelihood over the model parameters.

Question 5 on its own page

Question 6

+3 marksOne or more correct options

Which of the following statements is/are true?

Select all that apply.

A
EM algorithm converges to a global optimum.
B
Lloyd’s algorithm converges to a local optimum.
C
Poor initialization does not affect the clusters produced by EM algorithm.
D
Poor initialization leads to a slower convergence of Lloyd’s algorithm.
E
Lloyd’s algorithm is sensitive to the number of clusters (k).

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B
Lloyd’s algorithm converges to a local optimum.
D
Poor initialization leads to a slower convergence of Lloyd’s algorithm.
E
Lloyd’s algorithm is sensitive to the number of clusters (k).

Question 6 on its own page

Question 7

+3 marksOne or more correct options

For a linear regression problem, which among the following are true? Select all that apply.

Select all that apply.

A
Gradient descent will converge to a global minimum.
B
Gradient descent will converge to a local minimum (which is not a global minimum).
C
Stochastic gradient descent is guaranteed to converge to a global minimum.
D
Gradient descent will take lesser number of iterations to converge to a local minimum than stochastic gradient descent.

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Correct answers

A
Gradient descent will converge to a global minimum.
D
Gradient descent will take lesser number of iterations to converge to a local minimum than stochastic gradient descent.

Question 7 on its own page

Question 8

+3 marksNumerical answer

Show answer

Correct answer: 0.125 (accepted within ±0.005)

Question 8 on its own page

Question 9

+3 marksNumerical answer

Let $X_1, X_2, \ldots, X_n$ be an i.i.d. sample drawn from a Pareto distribution with the following probability density function:

$f(x \mid \theta) = \theta x^{-\theta-1}, \quad x \geq 1, \; \theta > 1.$

You are given the following sample of 5 observations:

$X = \{2.5,\ 1.8,\ 3.2,\ 2.0,\ 2.7\}.$

What is the maximum likelihood estimator of $\theta$ for the above sample? Enter the answer correct to three decimal places.

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Correct answer: 1.1485 (accepted within ±0.0015)

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Question 10

+2 marksNumerical answer

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Correct answer: 5250

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Question 11

+2 marksOne correct option

Based on the above data, answer the given subquestions.

Which of the following represents the covariance matrix C?

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Correct answer

Question 11 on its own page

Question 12

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

What is the variance of the dataset along the first principal component?

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Correct answer: 1

Question 12 on its own page

Question 13

+3 marksOne correct option

Based on the above data, answer the given subquestions.

Find the residues after projecting the data points onto the first principal component.

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Correct answer

Question 13 on its own page

Question 14

+3 marksOne correct option

Consider the following dataset with feature vector $\mathbf{x}$ and label $y$ :

$\mathbf{x}$	$y$
$\begin{bmatrix} 4 & 1 \end{bmatrix}^T$	2
$\begin{bmatrix} 2 & 8 \end{bmatrix}^T$	$-14$
$\begin{bmatrix} 1 & 4 \end{bmatrix}^T$	$-7$
$\begin{bmatrix} 3 & 2 \end{bmatrix}^T$	$-1$

Based on the above data, answer the given subquestions.

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Correct answer

Question 14 on its own page

Question 15

+1 markNumerical answer

Consider the following dataset with feature vector $\mathbf{x}$ and label $y$ :

$\mathbf{x}$	$y$
$\begin{bmatrix} 4 & 1 \end{bmatrix}^T$	2
$\begin{bmatrix} 2 & 8 \end{bmatrix}^T$	$-14$
$\begin{bmatrix} 1 & 4 \end{bmatrix}^T$	$-7$
$\begin{bmatrix} 3 & 2 \end{bmatrix}^T$	$-1$

Based on the above data, answer the given subquestions.

Find the mean squared error for the training dataset.

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Correct answer: 0

Question 15 on its own page