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IIT M FOUNDATION AN EXAM QDF2 26 Oct - 26 Oct 2025

MLT Quiz 114 questions40 marks6 SA · 5 MSQ · 3 MCQFoundation / afternoon / QDF2

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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 essential for the Diploma-Level Machine Learning Techniques exam, based on the question paper. The focus is on clustering (K-means, Lloyd’s algorithm), PCA, Bayesian inference, and kernel methods.

1. Core Topics & Key Concepts

A. Clustering (K-means & Lloyd’s Algorithm)

Lloyd’s Algorithm (K-means):
- Steps:
  1. Initialize cluster centers (e.g., randomly or via K-means++).
  2. Assign points to the nearest center (Euclidean distance).
  3. Recompute centers as the mean of assigned points.
  4. Repeat until convergence (centers stabilize or max iterations reached).
- Properties:
  - Convergence: Always converges to a local minimum (not necessarily global). (Question 2: Option A is true; Option B is false because it *always converges, though the solution may be suboptimal.)*
  - Sensitivity to Outliers: Yes (Option C is true).
  - Optimality: Not guaranteed (Option D is false).
  - Initialization Matters: Different initializations can lead to different results.
- K-means++ Initialization:
  - First center chosen uniformly at random.
  - Subsequent centers chosen with probability proportional to distance squared from existing centers.
  - Reduces variance in clustering quality.
  (Question 4: Probability of selecting (2,2) is higher due to distance from (0,0).)

Convergence Criteria:
- Stops when cluster assignments or centers change negligibly (e.g., below a threshold).
- (Question 14: Final cluster means depend on initialization.)

Tie-Breaking in K-means:
- Affects final clusters if points are equidistant to multiple centers.
- (Question 4: Option D is true.)

B. Principal Component Analysis (PCA)

Key Idea: Linear dimensionality reduction by projecting data onto orthogonal axes (principal components) that maximize variance.
Steps:
1. Standardize data (mean=0, variance=1).
2. Compute covariance matrix: $\Sigma = \frac{1}{n-1} X^T X$ .
3. Eigen decomposition: $\Sigma = V \Lambda V^T$ , where:
  - $V$ : Matrix of eigenvectors (principal components, sorted by eigenvalues).
  - $\Lambda$ : Diagonal matrix of eigenvalues (variances along PCs).
4. Project data onto top- $k$ eigenvectors: $X_{\text{new}} = X V_k$ .
Variance Explained:
- Total variance = sum of all eigenvalues.
- Variance of $i$ -th PC = $\lambda_i$ .
- (Question 3: For 6 points in 2D, covariance matrix has 2 non-zero eigenvalues. Top 6 PCs capture total variance only if all eigenvalues are non-zero.)
- (Questions 10–11: Variances along PC1 and PC2 are 4.5 and 0.5, respectively.)
Reconstruction:
- Zero residue only if all PCs are used (Question 3: Option C is false).
Covariance Matrix:
- For centered data $X$ , $\Sigma = \frac{1}{n-1} X^T X$ .
- (Question 9: Compute covariance from data.)

C. Bayesian Inference (Beta Distribution)

Beta Distribution:
- Prior for binomial proportions: $\text{Beta}(\alpha, \beta)$ .
- Posterior after observing $k$ successes in $n$ trials: $\text{Beta}(\alpha + k, \beta + n - k)$ .
- Posterior Mean: $\frac{\alpha + k}{\alpha + \beta + n}$ .
- *(Question 5:
  - For $\text{Beta}(1,1)$ (uniform prior) and data, posterior mean ≈ 0.039.
  - For $\text{Beta}(0.5,0.5)$ , posterior mean ≈ 0.034.)*

D. Kernel Methods

Kernel Trick: Map data to higher-dimensional space via $\phi(x)$ , compute dot products efficiently using kernels $K(x_i, x_j) = \phi(x_i)^T \phi(x_j)$ .
Valid Kernels (must satisfy Mercer’s condition):
- Linear: $K(x_i, x_j) = x_i^T x_j$ .
- Polynomial: $K(x_i, x_j) = (x_i^T x_j + c)^d$ .
- Gaussian (RBF): $K(x_i, x_j) = \exp(-\gamma \|x_i - x_j\|^2)$ .
- Sigmoid: $K(x_i, x_j) = \tanh(\alpha x_i^T x_j + c)$ .
- (Question 6: Options A, B, D are valid kernels.)

2. Key Equations

A. K-means

Objective: Minimize within-cluster sum of squares (WCSS):

\text{WCSS} = \sum_{i=1}^k \sum_{x \in C_i} \|x - \mu_i\|^2

where $\mu_i$ is the mean of cluster $C_i$ .

B. PCA

Covariance Matrix:

\Sigma = \frac{1}{n-1} X^T X \quad \text{(for centered } X\text{)}

Eigen Decomposition:

\Sigma = V \Lambda V^T

Variance of PC $i$ :

\text{Var}(PC_i) = \lambda_i

Total Variance:

\text{Total Variance} = \sum_{i=1}^d \lambda_i

C. Bayesian Posterior (Beta)

Posterior Mean:

\text{Mean} = \frac{\alpha + k}{\alpha + \beta + n}

D. Kernel Functions

Linear Kernel:

K(x_i, x_j) = x_i^T x_j

Gaussian (RBF) Kernel:

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

Graph 2: Bayesian Inference & Kernels

Graph 3: Workflow for K-means and PCA

4. Exam Tips

K-means:
- Know how initialization (random vs. K-means++) affects convergence.
- Understand why it’s sensitive to outliers (uses Euclidean distance).
- Remember it converges to a local minimum, not necessarily global.

PCA:
- Always standardize data first.
- Eigenvalues = variances; eigenvectors = principal components.
- Total variance = sum of all eigenvalues.

Bayesian Inference:
- For Beta prior, posterior mean is $\frac{\alpha + k}{\alpha + \beta + n}$ .
- Uniform prior = Beta(1,1).

Kernels:
- Valid kernels must satisfy Mercer’s condition (symmetric, positive semi-definite).
- Common kernels: linear, polynomial, Gaussian, sigmoid.