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IIT M IMPROVEMENT AN EXAM QIE2 26 Oct - 26 Oct 2025

MLT Quiz 114 questions40 marks6 SA · 5 MSQ · 3 MCQImprovement / afternoon / QIE2

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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 (K-Means, K-Means++, Lloyd’s Algorithm)

Lloyd’s Algorithm (K-Means):
- Iterative algorithm to minimize within-cluster variance.
- Convergence: Always converges (monotonically decreases objective function), but may converge to local optima.
- Sensitivity: Sensitive to outliers (since it uses Euclidean distance).
- Initialization: K-Means++ improves initialization by selecting centers based on distance-weighted probabilities.
- Equation (Objective Function):

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

K-Means++ Initialization Score:

S(x) = \min_{j=1,\dots,l-1} \|x - \mu_j^0\|^2

Convergence Behavior:
- May converge to suboptimal solutions depending on initialization.
- Not guaranteed to find the global optimum.

B. Principal Component Analysis (PCA)

Key Idea: Orthogonal transformation to convert correlated variables into linearly uncorrelated principal components (PCs).
Eigenvalues & Variance:
- Eigenvalues of the covariance matrix represent variance along PCs.
- If data lies in a k-dimensional subspace, only k eigenvalues are non-zero.
- Total variance = Sum of all eigenvalues.
- Proportion of variance explained by PC_i:

\text{Variance}_i = \frac{\lambda_i}{\sum_{j=1}^{d} \lambda_j}

Reconstruction Error:
- If k components capture all variance, reconstruction error = 0.
- Residue (error) after k components:

\text{Error} = \sum_{i=k+1}^{d} \lambda_i

Mean-Centered Data:
- PCA is applied to centered data (subtract mean from each feature).

C. Bayesian Statistics (Beta Distribution, Posterior Mean)

Beta Distribution as Prior for Proportion (θ):
- Prior: $\theta \sim \text{Beta}(a, b)$
- Posterior (after observing k successes in n trials):

\theta \mid \text{data} \sim \text{Beta}(a + k, b + n - k)

Posterior Mean (Estimate for θ):

E[\theta] = \frac{a + k}{a + b + n}

Special Cases:
- Uniform Prior (Beta(1,1)): Uninformative prior.
- Jeffreys Prior (Beta(0.5,0.5)): Common for proportions.

D. Kernel Methods (Valid Kernel Functions)

Kernel Trick: Maps data into a higher-dimensional space where it becomes linearly separable.
Valid Kernels (Must satisfy Mercer’s Condition):
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 Example: $K(x, y) = |x - y|$ (not positive semi-definite).

E. Expectation-Maximization (EM) 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 \mid \mu_k, \sigma_k^2)

where:

$\pi_k$ : Mixing coefficient (prior probability of cluster k).
$\mathcal{N}(x_i \mid \mu_k, \sigma_k^2)$ : Gaussian PDF for cluster k.
- E-Step (Posterior Probability):

\lambda_{ik} = \frac{\pi_k \cdot \mathcal{N}(x_i \mid \mu_k, \sigma_k^2)}{\sum_{j=1}^{K} \pi_j \cdot \mathcal{N}(x_i \mid \mu_j, \sigma_j^2)}

M-Step (Update Parameters):

\mu_k = \frac{\sum_{i=1}^{n} \lambda_{ik} x_i}{\sum_{i=1}^{n} \lambda_{ik}}, \quad \sigma_k^2 = \frac{\sum_{i=1}^{n} \lambda_{ik} \|x_i - \mu_k\|^2}{\sum_{i=1}^{n} \lambda_{ik}}

F. Covariance Matrix & PCA on 2D Data

Covariance Matrix (Σ) for 2D Data:

\Sigma = \begin{pmatrix} \text{Var}(X_1) & \text{Cov}(X_1, X_2) \\ \text{Cov}(X_2, X_1) & \text{Var}(X_2) \end{pmatrix}

Eigenvalues & Principal Components:
- Larger eigenvalue → 1st PC (max variance direction).
- Smaller eigenvalue → 2nd PC (orthogonal, next max variance).

2. Key Equations Summary

Topic	Key Equation
K-Means Objective	$J = \sum_{i=1}^{n} \min_{\mu_j} \\|x_i - \mu_j\\|^2$
K-Means++ Score	$S(x) = \min_{j} \\|x - \mu_j^0\\|^2$
PCA Variance	$\text{Variance}_i = \frac{\lambda_i}{\sum \lambda_j}$
Beta Posterior Mean	$E[\theta] = \frac{a + k}{a + b + n}$
GMM Likelihood	$L(\theta) = \prod_{i} \sum_{k} \pi_k \mathcal{N}(x_i \mid \mu_k, \sigma_k)$
EM Posterior (λ)	$\lambda_{ik} = \frac{\pi_k \mathcal{N}(x_i \mid \mu_k, \sigma_k)}{\sum_j \pi_j \mathcal{N}(x_i \mid \mu_j, \sigma_j)}$

Graph 2: PCA & Dimensionality Reduction

Graph 3: Bayesian Statistics & GMM

Graph 4: Kernel Methods

4. Exam Tips

K-Means:
- Know when it converges (always) vs. finds global optimum (not guaranteed).
- Understand K-Means++ initialization (probability ∝ distance²).

PCA:
- If data lies in a k-dimensional subspace, only k eigenvalues are non-zero.
- Reconstruction error = sum of discarded eigenvalues.

Bayesian Statistics:
- Beta(1,1) = Uniform prior, Beta(0.5,0.5) = Jeffreys prior.
- Posterior mean = $(a + k)/(a + b + n)$ .

GMM & EM:
- Likelihood involves sum over clusters (latent variables).
- λ (responsibility) is the posterior probability of a point belonging to a cluster.

Kernel Functions:
- Gaussian, Polynomial, Laplacian are valid.
- Absolute difference (|x-y|) is invalid.

Covariance Matrix:
- For centered data, covariance matrix is $X^T X / (n-1)$ .
- Eigenvalues give variance along PCs.

This summary covers all key concepts, equations, and relationships needed for the exam. The Mermaid graphs help visualize connections between topics. Focus on equations (especially for PCA, Bayesian, GMM) and algorithmic behavior (K-Means, EM).