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IIT M IMPROVEMENT AN EXAM QIM4 26 Oct - 26 Oct 2025

MLT Quiz 114 questions40 marks6 SA · 5 MSQ · 3 MCQImprovement / afternoon / QIM4

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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 other initialization.
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, followed by Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Key Concepts

A. Clustering (K-Means & K-Means++)

Lloyd’s Algorithm (K-Means):
- Convergence: Always converges (monotonically decreases objective function).
- Sensitivity: Sensitive to outliers (uses Euclidean distance).
- Optimality: May converge to local optima (not guaranteed global optimum).
- Initialization: Performance depends on initial centroids.

K-Means++ Initialization:
- Goal: Spread out initial centroids to improve convergence.
- Score Function: $S(x) = \min_{j} \|x - \mu_j\|^2$ (probability ∝ $S(x)$ ).
- Convergence: Faster than random initialization.

Key Equations:
- Centroid Update: $\mu_k = \frac{1}{|C_k|} \sum_{x \in C_k} x$
- Objective Function: $J = \sum_{k=1}^K \sum_{x \in C_k} \|x - \mu_k\|^2$

B. Principal Component Analysis (PCA)

Goal: Dimensionality reduction via orthogonal transformation.
Key Ideas:
- Projects data onto directions (principal components) of maximum variance.
- Eigenvalues of covariance matrix $\Sigma$ represent variance along PCs.
- Reconstruction Error: Zero if $k = \text{rank}(X)$ .

Key Equations:
- Covariance Matrix: $\Sigma = \frac{1}{n} X^T X$ (for centered data).
- Variance Explained: $\text{Proportion} = \frac{\lambda_i}{\sum_j \lambda_j}$
- Reconstruction: $X \approx U_k \Sigma_k V_k^T$ (top- $k$ SVD).

Properties:
- If data lies in a $d$ -dimensional subspace, only $d$ eigenvalues are non-zero.
- Total Variance: Sum of all eigenvalues = sum of diagonal entries of $\Sigma$ .

C. Bayesian Statistics (Beta-Binomial Model)

Prior: $\theta \sim \text{Beta}(a, b)$ (conjugate prior for Binomial).
Posterior: $\theta | \text{data} \sim \text{Beta}(a + \text{successes}, b + \text{failures})$ .
Posterior Mean (Estimate for $\theta$ ):

E[\theta] = \frac{a + \text{successes}}{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 = $\frac{1 + 3}{1 + 1 + 100} = 0.039$ .

D. Kernel Methods

Valid Kernels (Mercer’s Condition):
- Gaussian (RBF): $K(x,y) = \exp\left(-\frac{\|x-y\|^2}{2\sigma^2}\right)$ .
- Polynomial: $K(x,y) = (x \cdot y + c)^d$ .
- Laplacian: $K(x,y) = \exp\left(-\frac{\|x-y\|}{\sigma}\right)$ .
- Invalid: $K(x,y) = |x-y|$ (not positive semi-definite).

E. Gaussian Mixture Models (GMM) & EM Algorithm

Likelihood Function:

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

E-Step: Compute responsibilities ( $\lambda_{ik}$ ):

\lambda_{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:

\pi_k = \frac{1}{n} \sum_{i=1}^n \lambda_{ik}, \quad \mu_k = \frac{\sum_i \lambda_{ik} x_i}{\sum_i \lambda_{ik}}

F. Linear Algebra for PCA

Rank & Linear Independence:
- If columns of $X$ are linearly dependent, $\text{rank}(X) < d$ .
- Example: $c_4 = 2c_1 - c_2$ → $\text{rank} = 3$ (only 3 independent columns).
Reconstruction Error: Zero if $k \geq \text{rank}(X)$ .

Graph 2: Bayesian & Kernel Methods

Graph 3: Linear Algebra for PCA

3. Key Equations Summary

Topic	Equation	Notes
K-Means Objective	$J = \sum_{k=1}^K \sum_{x \in C_k} \\|x - \mu_k\\|^2$	Minimize intra-cluster variance.
PCA Variance	$\text{Variance PC}_i = \lambda_i$	$\lambda_i$ : Eigenvalue of $\Sigma$ .
Posterior Mean (Beta)	$E[\theta] = \frac{a + \text{successes}}{a + b + n}$	Bayesian update for Binomial.
Gaussian Kernel	$K(x,y) = \exp\left(-\frac{\\|x-y\\|^2}{2\sigma^2}\right)$	Valid kernel (RBF).
GMM Likelihood	$L(\theta) = \prod_i \sum_k \pi_k \mathcal{N}(x_i \| \mu_k, \Sigma_k)$	Log-likelihood for EM.
E-Step (EM)	$\lambda_{ik} = \frac{\pi_k \mathcal{N}(x_i \| \mu_k, \Sigma_k)}{\sum_j \pi_j \mathcal{N}(x_i \| \mu_j, \Sigma_j)}$	Responsibilities.

4. Exam Tips

PCA:
- If data lies in a $d$ -dimensional subspace, only $d$ eigenvalues are non-zero.
- Variance explained = $\frac{\lambda_i}{\sum \lambda_j}$ .
- Reconstruction error = 0 if $k \geq \text{rank}(X)$ .

K-Means:
- Converges always (but may be suboptimal).
- K-Means++ improves initialization by weighting points by $S(x)$ .

Bayesian:
- Beta prior + Binomial data → Beta posterior.
- Posterior mean = $\frac{a + \text{successes}}{a + b + n}$ .

Kernels:
- RBF, Polynomial, Laplacian are valid.
- $|x-y|$ is not a valid kernel.

EM Algorithm:
- E-Step: Compute $\lambda_{ik}$ (responsibilities).
- M-Step: Update $\pi_k, \mu_k, \Sigma_k$ .

Linear Algebra:
- Rank = number of linearly independent columns/rows.
- If $c_4 = 2c_1 - c_2$ , the rank reduces by 1.