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IIT M IMPROVEMENT AN EXAM QID2 26 Oct - 26 Oct 2025

MLT Quiz 114 questions40 marks6 SA · 5 MSQ · 3 MCQImprovement / afternoon / QID2

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Q1MSQ2 marksselect all that apply

Which among the following is true with respect to the Lloyds’ algorithm?

It always converges.
It may fail to converge for an initialization while it may converge for otherinitialization.
It is sensitive to outliers.
The solution produced is always the optimal.

Q2MSQ4 marksselect all that apply

There will be 50 non-zero eigenvalues of the covariance matrix.
There will be 6 non-zero eigenvalues of the covariance matrix.
Using the first 5 principal components, the dataset can be reconstructed withzero residue.
The variance captured by the top 6 principal components equals the totalvariance of the dataset.

Q3MSQ4 marksselect all that apply

The points (0, 2), (2, 0), and (2, 2) have the equal probability of being chosen asthe second cluster center.
The probability of selecting (2, 2) as the second cluster center is twice that ofselecting (0, 2).
If (2, 2) is chosen as the second cluster center, the algorithm will alwaysconverge in one iteration.
The tie-breaking rule will affect the final cluster centers.

Q4MSQ4 marksselect all that apply

Posterior mean for Beta(1, 1) is 0.039.
Posterior mean for Beta(1, 1) is 0.03.
Posterior mean for Beta(0.5, 0.5) is 0.025.
Posterior mean for Beta(0.5, 0.5) is 0.034.

Q5MSQ3 marksselect all that apply

Which of the following are valid kernel functions? Select all that apply.

Q6SA3 marks

Your numeric answer:

Q7SA3 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q8MCQ2 marks

What is the covariance matrix of the dataset?

Q9SA2 marks

If standard PCA is performed on this dataset, what is the variance along the first principal component? Enter the answer correct to one decimal place.

Your numeric answer:

Q10SA2 marks

If standard PCA is performed on this dataset, what is the variance along the second principal component? Enter the answer correct to one decimal place.

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q11MCQ3 marks

Q12SA3 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q13MCQ3 marks

Perform K-means clustering with K = 2 and initial cluster centers at (−2, 3) and (2,−1). What are the final means of clusters after convergence?

Q14SA2 marks

Suppose we introduce a new data point (0, 0) to the existing dataset and perform K means with the same initial cluster centers. How many iterations would it require now for convergence?

Your numeric answer:

Answer key

Q1 (MSQ, 2 marks): A, C
Q2 (MSQ, 4 marks): B, D
Q3 (MSQ, 4 marks): B, C, D
Q4 (MSQ, 4 marks): A, D
Q5 (MSQ, 3 marks): A, B, D
Q6 (SA, 3 marks): 3
Q7 (SA, 3 marks): 0.45 to 0.50
Q8 (MCQ, 2 marks): B
Q9 (SA, 2 marks): 4.5
Q10 (SA, 2 marks): 0.5
Q11 (MCQ, 3 marks): B
Q12 (SA, 3 marks): 0.64 to 0.68
Q13 (MCQ, 3 marks): A
Q14 (SA, 2 marks): 1

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations for the Diploma-Level Machine Learning Techniques Exam, along with Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Key Concepts

A. Clustering Algorithms

1. K-Means & K-Means++

Lloyd’s Algorithm (K-Means):
- Convergence: Always converges (monotonically decreases objective function, but may reach local minima).
- Sensitivity: Sensitive to outliers and initialization.
- Optimality: Does not guarantee global optimum (depends on initialization).
- Objective Function:

J = \sum_{i=1}^{n} \min_{\mu_j \in C} \|x_i - \mu_j\|^2

K-Means++ Initialization:
- First center chosen uniformly at random.
- Subsequent centers chosen with probability proportional to $D(x)^2$ (squared distance to nearest center).
- Score for datapoint $x$ :

S(x) = \min_{j=1,\dots,l-1} \|x - \mu_j\|^2

2. EM Algorithm for Gaussian Mixture Models (GMM)

Likelihood Function for GMM:

L(\theta) = \prod_{i=1}^{n} \sum_{k=1}^{K} \pi_k \cdot \mathcal{N}(x_i | \mu_k, \Sigma_k)

where:

$\pi_k$ : Mixing coefficient (prior probability of cluster $k$ ).
$\mathcal{N}(x | \mu, \Sigma)$ : Gaussian PDF.
- E-Step: Compute responsibilities ( $\gamma_{ik}$ ):

\gamma_{ik} = \frac{\pi_k \mathcal{N}(x_i | \mu_k, \Sigma_k)}{\sum_{j=1}^{K} \pi_j \mathcal{N}(x_i | \mu_j, \Sigma_j)}

M-Step: Update parameters:

\mu_k = \frac{\sum_{i=1}^{n} \gamma_{ik} x_i}{\sum_{i=1}^{n} \gamma_{ik}}, \quad \Sigma_k = \frac{\sum_{i=1}^{n} \gamma_{ik} (x_i - \mu_k)(x_i - \mu_k)^T}{\sum_{i=1}^{n} \gamma_{ik}}

B. Dimensionality Reduction (PCA)

1. Principal Component Analysis (PCA)

Goal: Find orthogonal axes (principal components) that maximize variance.
Steps:
1. Center the data (subtract mean).
2. Compute covariance matrix $S = \frac{1}{n-1} X^T X$ .
3. Eigendecomposition: $S = U \Lambda U^T$ , where:
  - $U$ : Matrix of eigenvectors (principal components).
  - $\Lambda$ : Diagonal matrix of eigenvalues (variances).
4. Project data onto top- $k$ eigenvectors.
Variance Explained:

\text{Proportion of variance} = \frac{\lambda_i}{\sum_{j=1}^{d} \lambda_j}

Reconstruction Error (Residue):
- If data lies in a $k$ -dimensional subspace, the first $k$ PCs capture all variance.
- Eigenvalues = 0 for dimensions beyond intrinsic dimensionality.

2. Key Properties

Covariance Matrix Rank:
- If data lies in a $k$ -dimensional subspace, covariance matrix has $k$ non-zero eigenvalues.
Example:
- If all points lie in a 6D subspace in $\mathbb{R}^{50}$ , PCA yields:
  - 6 non-zero eigenvalues.
  - First 6 PCs capture 100% variance.

C. Bayesian Statistics

1. Beta-Binomial Model for Proportions

Prior: $\theta \sim \text{Beta}(a, b)$
Likelihood: $X \sim \text{Binomial}(n, \theta)$
Posterior: $\theta | X \sim \text{Beta}(a + x, b + n - x)$
Posterior Mean (Estimate for $\theta$ ):

\mathbb{E}[\theta | X] = \frac{a + x}{a + b + n}

Example:
- Prior: $\text{Beta}(1,1)$ (uniform) or $\text{Beta}(0.5, 0.5)$ .
- Data: 3 defects in 100 samples.
- Posterior Mean:
  - For $\text{Beta}(1,1)$ : $\frac{1 + 3}{1 + 1 + 100} = 0.039$
  - For $\text{Beta}(0.5,0.5)$ : $\frac{0.5 + 3}{0.5 + 0.5 + 100} \approx 0.034$

D. Kernel Methods

1. Valid Kernel Functions

A function $K(x, y)$ is a valid kernel if it satisfies Mercer’s condition (positive semi-definite Gram matrix).

Examples:
1. Gaussian (RBF) Kernel:

K(x, y) = \exp\left(-\frac{\|x - y\|^2}{2\sigma^2}\right)

Polynomial Kernel:

K(x, y) = (x \cdot y + c)^d

Laplacian Kernel:

K(x, y) = \exp\left(-\frac{\|x - y\|}{\sigma}\right)

Invalid Kernel:
- $K(x, y) = |x - y|$ (not PSD).

2. Key Equations Summary

Topic	Equation
K-Means Objective	$J = \sum_{i=1}^{n} \min_{\mu_j} \\|x_i - \mu_j\\|^2$
K-Means++ Score	$S(x) = \min_{j} \\|x - \mu_j\\|^2$
PCA Variance	$\text{Var}(PC_i) = \lambda_i$
PCA Proportion	$\frac{\lambda_i}{\sum \lambda_j}$
GMM Likelihood	$L(\theta) = \prod_{i} \sum_{k} \pi_k \mathcal{N}(x_i \| \mu_k, \Sigma_k)$
Beta Posterior Mean	$\mathbb{E}[\theta \| X] = \frac{a + x}{a + b + n}$
Gaussian Kernel	$K(x, y) = \exp\left(-\frac{\\|x - y\\|^2}{2\sigma^2}\right)$

Graph 2: Dimensionality Reduction (PCA)

Graph 3: Bayesian Statistics & Kernels

4. Exam Tips

PCA:
- If data lies in a $k$ -dimensional subspace, first $k$ PCs capture all variance.
- Eigenvalues = 0 for redundant dimensions.
K-Means:
- Converges always but may not find the global optimum.
- K-Means++ improves initialization by weighting distances.
GMM:
- Likelihood involves sum over components (mixture model).
- EM alternates between E-step (expectation) and M-step (maximization).
Bayesian Estimation:
- Posterior mean is the estimate for $\theta$ .
- Beta prior is conjugate for Binomial likelihood.
Kernels:
- RBF, Polynomial, Laplacian are valid.
- Absolute difference is not a valid kernel.