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IIT M FOUNDATION DIPLOMA FN EXAM - 01 Sep 2024

MLT End Term30 questions50 marks14 SA · 13 MCQ · 3 MSQFoundation / Diploma / forenoon

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Q1MCQ3 marks

Q2MCQ3 marks

Q3MCQ3 marks

Match the characteristics given below with the appropriate ensemble technique. 1. Decision stumps 2. Deep decision trees 3. Parallel execution 4. Sequential execution

Bagging → (2), (3); Boosting → (1), (4)
Bagging → (1), (4); Boosting → (2), (3)
Bagging → (1), (3); Boosting → (2), (4)
Bagging → (2), (4); Boosting → (1), (3)

Q4MCQ3 marks

The predicted labels for x1 and x2 are 1 and −1 respectively.
The predicted labels for x1 and x2 are −1 and 1 respectively.
The predicted label for both data-points is 1.
The predicted label for both data-points is −1.

Q5MCQ3 marks

Q6MCQ3 marks

2x1 + x2 = 0
x1 − 2x2 = 0
x1 + x2 = 0
x1 + 2x2 = 0

Q7SA3 marks

Your numeric answer:

Q8SA3 marks

Your numeric answer:

Q9SA3 marks

Your numeric answer:

Q10MSQ3 marksselect all that apply

Q11MSQ3 marksselect all that apply

The dataset is linearly separable with a positive margin.
The perceptron algorithm will terminate after a finite number of iterationswhen trained on this dataset.
The dataset is linearly separable, but the margin may be zero.
The dataset is not linearly separable.

Q12MSQ3 marksselect all that apply

Comprehension passage - the questions below it refer to this

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

Q13MCQ1 mark

The weight vector output by the perceptron algorithm on this dataset can be expressed as a linear combination of the data-points where the coefficients of the linear combination are integers.

TRUE
FALSE

Q14MCQ1 mark

TRUE
FALSE

Q15MCQ1 mark

If a soft-margin SVM is trained on this dataset, the optimal weight vector it returns will be the same as the one returned by a hard-margin SVM, irrespective of the value of the hyperparameter C.

TRUE
FALSE

Comprehension passage - the questions below it refer to this

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

Q16SA0.5 marks

Your numeric answer:

Q17SA0.5 marks

Your numeric answer:

Q18SA0.5 marks

Your numeric answer:

Q19SA0.5 marks

Your numeric answer:

Q20SA0.5 marks

Your numeric answer:

Q21SA0.5 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q22SA0.5 marks

Your numeric answer:

Q23SA0.5 marks

Your numeric answer:

Q24SA0.5 marks

Your numeric answer:

Q25SA0.5 marks

Enter the number corresponding to the squared loss._______________

Your numeric answer:

Q26MCQ1 mark

Comprehension passage - the questions below it refer to this

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

Q27SA1 mark

How many learnable parameters does this network have? Ignore the biases in the computation.

Your numeric answer:

Q28MCQ1 mark

What is the most appropriate choice of activation function for the output layer if the binary cross-entropy loss is used?

Sigmoid
Linear
ReLU

Q29MCQ1 mark

ReLU
Sigmoid

Q30MCQ2 marks

Answer key

Q1 (MCQ, 3 marks): A
Q2 (MCQ, 3 marks): A
Q3 (MCQ, 3 marks): A
Q4 (MCQ, 3 marks): A
Q5 (MCQ, 3 marks): A
Q6 (MCQ, 3 marks): A
Q7 (SA, 3 marks): 0.4
Q8 (SA, 3 marks): 30
Q9 (SA, 3 marks): 0.26 to 0.37
Q10 (MSQ, 3 marks): A, B
Q11 (MSQ, 3 marks): A, B
Q12 (MSQ, 3 marks): A, B
Q13 (MCQ, 1 marks): A
Q14 (MCQ, 1 marks): B
Q15 (MCQ, 1 marks): B
Q16 (SA, 0.5 marks): 2
Q17 (SA, 0.5 marks): 1
Q18 (SA, 0.5 marks): 0
Q19 (SA, 0.5 marks): 3
Q20 (SA, 0.5 marks): 0
Q21 (SA, 0.5 marks): -1
Q22 (SA, 0.5 marks): 3
Q23 (SA, 0.5 marks): 2
Q24 (SA, 0.5 marks): 4
Q25 (SA, 0.5 marks): 1
Q26 (MCQ, 1 marks): A
Q27 (SA, 1 marks): 160
Q28 (MCQ, 1 marks): A
Q29 (MCQ, 1 marks): A
Q30 (MCQ, 2 marks): A

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 exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Concepts

1. Principal Component Analysis (PCA)

Key Idea: Dimensionality reduction by projecting data onto directions (principal components) that maximize variance.
Eigenvalues of Covariance Matrix:
- Represent variance along principal components.
- Total variance = sum of eigenvalues.
- Variance captured by top k PCs = (sum of top k eigenvalues) / (total variance).
Equation:

\text{Variance captured} = \frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^d \lambda_i}

(For Q2: $\frac{15 + 5}{15 + 5 + 5 + 0 + 0} = 0.8$ → 80%)

2. Regularization & Ridge Regression

Key Idea: Penalize large weights to prevent overfitting.
Loss Function:

L(\mathbf{w}) = \text{SSE} + \lambda \|\mathbf{w}\|_2^2

where $\text{SSE} = \sum (y_i - \hat{y}_i)^2$ .

Plot Interpretation (Q3):
- X-axis: $f(\lambda) = \log(\lambda)$ (common choice to visualize a wide range of $\lambda$ values).
- Y-axis: Regularized loss (SSE + penalty).
- Curves: Training vs. test loss (U-shaped, with test loss minimizing at optimal $\lambda$ ).

3. Ensemble Methods (Bagging vs. Boosting)

Technique	Base Learners	Execution	Key Idea
Bagging	Deep decision trees	Parallel	Reduce variance via bootstrapping.
Boosting	Decision stumps	Sequential	Reduce bias via weighted corrections.

(For Q4: Bagging → (2), (3); Boosting → (1), (4))

4. Support Vector Machines (SVM)

Hard-Margin SVM

Decision Boundary: $\mathbf{w}^T \mathbf{x} + b = 0$ .
Supporting Hyperplanes: $\mathbf{w}^T \mathbf{x} + b = \pm 1$ .
Margin: $\frac{2}{\|\mathbf{w}\|}$ .
Prediction: $\text{sign}(\mathbf{w}^T \mathbf{x} + b)$ . (For Q5: Points beyond $+1$ → class $+1$ ; beyond $-1$ → class $-1$ .)

Soft-Margin SVM

Slack Variables ( $\xi_i$ ): Allow margin violations.
Objective:

\min_{\mathbf{w}, b, \xi} \frac{1}{2} \|\mathbf{w}\|^2 + C \sum \xi_i

subject to $y_i (\mathbf{w}^T \mathbf{x}_i + b) \geq 1 - \xi_i$ .

KKT Conditions:
- $\alpha_i^* (y_i (\mathbf{w}^T \mathbf{x}_i + b) - 1 + \xi_i) = 0$ .
- If $\alpha_i^* > 0$ , the point is a support vector (on margin or violated).
(For Q15–Q22: Analyze $\xi_i$ and $\alpha_i$ based on hyperplane positions.)

Kernel Trick

Polynomial Kernel:

K(\mathbf{x}, \mathbf{x}') = (\mathbf{x}^T \mathbf{x}' + c)^p

Non-negative outputs: Only possible for even $p$ (e.g., $p = 2, 4$ ). (For Q11: A, B)

5. Naive Bayes

Class Conditional Independence:

p(x_1, x_2, \dots, x_d \mid y) = \prod_{i=1}^d p(x_i \mid y)

(For Q6: Option A is correct.)

6. Clustering (k-Means)

Cluster Boundary: Perpendicular bisector between centroids. (For Q7: Equation $2x_1 + x_2 = 0$ separates ■ and × clusters.)

7. Decision Trees & Information Gain

Entropy:

H(S) = -\sum p_i \log_2 p_i

Information Gain:

\text{IG} = H(\text{parent}) - \sum \left( \frac{|S_v|}{|S|} H(S_v) \right)

(For Q10: 0.311 bits.)

8. Maximum Likelihood Estimation (MLE)

Bernoulli MLE:

\hat{p} = \frac{\text{number of successes}}{\text{total trials}}

(For Q9: $\hat{p} = 0.25$ , successes = 10 → total trials = 40 → zeros = 30.)

9. Loss Functions in Classification

Loss	Equation	Curve Color (Q23–Q26)
Logistic	$\log(1 + e^{-z})$	Red (3)
Hinge (SVM)	$\max(0, 1 - z)$	Green (2)
Perceptron	$\max(0, -z)$	Blue (4)
Squared	$(1 - z)^2$	Yellow (1)

(For Q23–Q26: Match curves to losses.)

10. Neural Networks

Architecture

Parameters: Weights between layers (ignore biases).
- Input → Hidden-1: $5 \times 10 = 50$ .
- Hidden-1 → Hidden-2: $10 \times 10 = 100$ .
- Hidden-2 → Output: $10 \times 1 = 10$ .
- Total: $50 + 100 + 10 = 160$ .
(For Q28: 160.)

Activation Functions

Output Layer (Binary Classification):
- Sigmoid (for cross-entropy loss).
(For Q29: A.)
Hidden Layer:
- ReLU: $\max(0, z)$ (sparse activations, e.g., zeros in Q30).

2. SVM Concepts

3. Ensemble Methods

4. Loss Functions

5. Neural Network Parameters

Key Equations Summary

Topic	Equation
PCA Variance	$\text{Variance captured} = \frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^d \lambda_i}$
Ridge Regression	$L(\mathbf{w}) = \text{SSE} + \lambda \\|\mathbf{w}\\|_2^2$
SVM Margin	$\frac{2}{\\|\mathbf{w}\\|}$
Naive Bayes	$p(\mathbf{x} \mid y) = \prod_i p(x_i \mid y)$
Entropy	$H(S) = -\sum p_i \log_2 p_i$
Information Gain	$\text{IG} = H(\text{parent}) - \sum w_v H(S_v)$
Bernoulli MLE	$\hat{p} = \frac{\text{successes}}{\text{trials}}$
Logistic Loss	$\log(1 + e^{-z})$
Hinge Loss	$\max(0, 1 - z)$
Neural Net Parameters	$\sum_{l=1}^{L-1} n_l \times n_{l+1}$ (ignoring biases)