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IIT M DIPLOMA AN EXAM QDD4 26 Oct 2025 - 26 Oct 2025

MLT Quiz 114 questions40 marks6 SA · 5 MSQ · 3 MCQDiploma / afternoon / QDD4

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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 provided question paper. The exam focuses on clustering (K-means, Lloyd’s algorithm), dimensionality reduction (PCA), Bayesian inference, and kernel methods.

1. Core Topics & Key Concepts

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

Lloyd’s Algorithm (K-means):
- Iterative algorithm to partition data into K clusters by minimizing within-cluster variance.
- Steps:
  1. Initialize K cluster centers (e.g., randomly or via k-means++).
  2. Assign each point to the nearest center (using Euclidean distance).
  3. Recompute centers as the mean of assigned points.
  4. Repeat until convergence (centers stabilize or max iterations reached).
- Convergence:
  - Guaranteed to converge (monotonically decreases objective function), but may converge to local optima (depends on initialization).
  - Sensitive to outliers (outliers can skew cluster centers).
- Time Complexity: O(n·K·I·d), where n = samples, K = clusters, I = iterations, d = dimensions.
- Initialization Methods:
  - Random, k-means++, or heuristic (e.g., selecting points farthest from existing centers).

Probability of Cluster Center Selection:
- If using k-means++, the probability of selecting a point as a center is proportional to its squared distance from existing centers.
- Example: For points (0,2), (2,0), (2,2), the probability of selecting (2,2) may be higher if it’s farther from the first center.

Tie-Breaking in K-means:
- When points are equidistant to multiple centers, the tie-breaking rule (e.g., assign to the lower-index center) can affect final clusters.

B. Dimensionality Reduction (PCA)

Principal Component Analysis (PCA):
- Linear transformation to project data into a lower-dimensional space while maximizing variance.
- Steps:
  1. Standardize data (mean=0, variance=1).
  2. Compute covariance matrix Σ = XᵀX (for centered data).
  3. Compute eigenvalues (λ) and eigenvectors (v) of Σ.
  4. Sort eigenvalues in descending order; select top k eigenvectors (principal components).
  5. Project data: X_proj = X · v_k.
- Key Properties:
  - The covariance matrix is d×d (for d dimensions). For n samples, there are min(n-1, d) non-zero eigenvalues.
  - Total variance = sum of all eigenvalues = trace(Σ).
  - Variance captured by top k PCs = sum of top k eigenvalues.
  - Reconstruction error: Zero if k = d (no dimensionality reduction).
- Example:
  - If Σ has eigenvalues [4.5, 0.5, 0, ..., 0], the first PC captures 4.5 units of variance, and the second captures 0.5.

C. Bayesian Inference (Beta Distribution)

Beta Distribution:
- Prior/conjugate prior for binomial likelihood. Defined as:

\text{Beta}(\alpha, \beta) \sim \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)}, \quad B(\alpha, \beta) = \text{Beta function}

Posterior Mean: For a Beta(α, β) posterior with s successes and f failures:

\text{Posterior mean} = \frac{\alpha + s}{\alpha + \beta + s + f}

Example:
- For Beta(1,1) (uniform prior) with 1 success and 24 failures:

\text{Mean} = \frac{1 + 1}{1 + 1 + 1 + 24} = \frac{2}{27} \approx 0.039.

For Beta(0.5,0.5) with the same data:

\text{Mean} = \frac{0.5 + 1}{0.5 + 0.5 + 1 + 24} = \frac{1.5}{26} \approx 0.034.

D. Kernel Methods

Kernel Functions:
- Enable non-linear decision boundaries by mapping data to a higher-dimensional space.
- Valid Kernels must satisfy Mercer’s condition (positive semi-definite Gram matrix).
- Common Kernels:
  1. Linear: K(x, y) = xᵀy.
  2. Polynomial: K(x, y) = (xᵀy + c)ᵈ.
  3. Gaussian (RBF): K(x, y) = exp(-γ||x-y||²).
  4. Sigmoid: K(x, y) = tanh(κxᵀy + c).
- Example Valid Kernels (from Q6): Likely options A/B/D included linear, polynomial, or RBF kernels.

2. Key Equations

Topic	Equation	Notes
K-means Objective	$\arg\min_{C} \sum_{i=1}^{n} \min_{c \in C} \\|x_i - c\\|^2$	Minimize within-cluster sum of squares.
Covariance Matrix	$\Sigma = \frac{1}{n-1} X^T X$ (centered data)	Diagonal entries = variances.
PCA Projection	$X_{\text{proj}} = X \cdot v_k$	$v_k$ = top k eigenvectors.
Beta Posterior Mean	$\frac{\alpha + s}{\alpha + \beta + s + f}$	s = successes, f = failures.
Gaussian Kernel	$K(x, y) = \exp(-\gamma \\|x - y\\|^2)$	$\gamma$ controls flexibility.

Graph 2: Bayesian Inference & Kernels

4. Exam Tips

K-means:
- Remember it always converges but may not find the global optimum.
- Outliers pull cluster centers toward them.
- Initialization matters (k-means++ is better than random).

PCA:
- Number of non-zero eigenvalues = min(n-1, d).
- First PC direction = eigenvector of largest eigenvalue.
- Total variance = sum of all eigenvalues.

Bayesian Inference:
- For Beta(α, β), the posterior mean is a weighted average of prior and data.
- Beta(1,1) is a uniform prior.

Kernels:
- Linear kernel is a dot product.
- RBF kernel introduces non-linearity via Gaussian similarity.