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IIT M DIPLOMA FN EXAM QDD2 13 Apr 2025 - 13 Apr 2025

MLF End Term14 questions40 marks8 MSQ · 5 SA · 1 MCQDiploma / forenoon / QDD2

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Q1MCQ3 marks

Q2MSQ3 marksselect all that apply

Which of the following is true about the singular value decomposition (SVD)?

Q3MSQ3 marksselect all that apply

In the context of Principal Component Analysis (PCA), which of the following statements is/are correct?

PCA projects the original data onto a lower-dimensional subspace thatcaptures the maximum variance.
The eigenvectors of the covariance matrix correspond to the directions ofmaximum variance.
The principal components form a basis that is not necessarily orthogonal.
The principal components obtained through PCA are always uncorrelated.

Q4MSQ3 marksselect all that apply

Q5MSQ3 marksselect all that apply

Which of the following statements about convex sets are true?

Q6MSQ3 marksselect all that apply

Which of the following options is/are true?

Q7MSQ4 marksselect all that apply

Q8SA3 marks

Your numeric answer:

Q9SA3 marks

Your numeric answer:

Q10SA2 marks

Your numeric answer:

Q11SA2 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

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

Q12MSQ2 marksselect all that apply

Which of the following statements correctly represents the Karush-Kuhn-Tucker (KKT) conditions for the given optimization problem?

Q13MSQ3 marksselect all that apply

Which of the following statements are correct based on the KKT conditions and the solution to the optimization problem?

Q14SA3 marks

Your numeric answer:

Answer key

Q1 (MCQ, 3 marks): A
Q2 (MSQ, 3 marks): A, B, C
Q3 (MSQ, 3 marks): A, B, D
Q4 (MSQ, 3 marks): C, D
Q5 (MSQ, 3 marks): A, B, C
Q6 (MSQ, 3 marks): A, C
Q7 (MSQ, 4 marks): A, B
Q8 (SA, 3 marks): 4
Q9 (SA, 3 marks): 0.5
Q10 (SA, 2 marks): 0.5
Q11 (SA, 2 marks): 1
Q12 (MSQ, 2 marks): A, C
Q13 (MSQ, 3 marks): A, C
Q14 (SA, 3 marks): 0.25

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 & Key Concepts

1. Probability & Statistics

Key Concepts:

Joint Probability Density Function (PDF):
- Transformation of random variables (e.g., $U = X + Y$ , $V = X - Y$ ).
- Jacobian determinant for change of variables.
- Support/range of transformed variables.
Expected Value & Linearity:
- $E[X_1 + X_2 + X_3] = E[X_1] + E[X_2] + E[X_3]$ .
- Expected values for common distributions:
  - Uniform( $a, b$ ): $E[X] = \frac{a+b}{2}$ .
  - Poisson( $\lambda$ ): $E[X] = \lambda$ .
  - Exponential( $\lambda$ ): $E[X] = \frac{1}{\lambda}$ .
Inequalities:
- Markov’s Inequality: $P(X \geq a) \leq \frac{E[X]}{a}$ .
- Chebyshev’s Inequality (not in questions but related).
Cumulative Distribution Function (CDF):
- For Uniform( $a, b$ ): $F_X(x) = \frac{x - a}{b - a}$ .

Key Equations:

Joint PDF Transformation:

f_{U,V}(u,v) = f_{X,Y}(x(u,v), y(u,v)) \cdot |J|

where $J$ is the Jacobian determinant:

J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}

Markov’s Inequality:

P(X \geq a) \leq \frac{E[X]}{a}

CDF for Uniform Distribution:

F_X(x) = \frac{x - a}{b - a} \quad \text{for } a \leq x \leq b

2. Linear Algebra & Matrix Decompositions

Key Concepts:

Singular Value Decomposition (SVD):
- For matrix $A \in \mathbb{R}^{m \times n}$ :
  - $A = U \Sigma V^T$ , where:
    - $U$ : Left singular vectors (eigenvectors of $AA^T$ ).
    - $V$ : Right singular vectors (eigenvectors of $A^T A$ ).
    - $\Sigma$ : Diagonal matrix of singular values (square roots of eigenvalues of $A^T A$ or $AA^T$ ).
  - Rank of $A$ = Number of non-zero singular values.
  - SVD is not unique for square matrices (depends on sign flips in $U$ and $V$ ).
Eigenvalues/Eigenvectors:
- Used in PCA and SVD.
- $Av = \lambda v$ .

Key Equations:

SVD Relationships:

A = U \Sigma V^T, \quad A^T A = V \Sigma^T \Sigma V^T, \quad AA^T = U \Sigma \Sigma^T U^T

Singular Values:

\sigma_i = \sqrt{\lambda_i(A^T A)}

3. Principal Component Analysis (PCA)

Key Concepts:

Goal: Project data onto a lower-dimensional subspace capturing maximum variance.
Steps:
1. Center the data (subtract mean).
2. Compute covariance matrix $\Sigma$ .
3. Eigenvalue decomposition: $\Sigma = V \Lambda V^T$ .
4. Principal components = eigenvectors sorted by eigenvalues (descending).
Properties:
- Principal components are orthogonal.
- Uncorrelated (if data is centered).
- First PC captures the most variance.

Key Equations:

Covariance Matrix:

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

Projection onto PCs:

Y = X W \quad \text{where } W = [v_1, v_2, \dots, v_k]

4. Convex Optimization

Key Concepts:

Convex Sets:
- A set $C$ is convex if $\forall x, y \in C$ , $\theta x + (1-\theta)y \in C$ for $\theta \in [0,1]$ .
- Examples: Lines, hyperplanes, intersections of convex sets.
Convex Functions:
- $f$ is convex if:

f(\theta x + (1-\theta)y) \leq \theta f(x) + (1-\theta)f(y)

Tests for Convexity:
- Hessian $\nabla^2 f \succeq 0$ (positive semidefinite).
- First-order condition: $f(y) \geq f(x) + \nabla f(x)^T (y - x)$ .
Examples: $e^x$ , $x^2$ , $-\ln x$ (if domain is convex).
- Gradient Descent:
Update rule:

x_{k+1} = x_k - \eta \nabla f(x_k)

Converges to minimum for convex $f$ with appropriate $\eta$ .
- Karush-Kuhn-Tucker (KKT) Conditions:
Necessary for optimality in constrained optimization.
For problem:

\min f(x) \quad \text{s.t.} \quad g_i(x) \leq 0, h_j(x) = 0

KKT conditions:

Stationarity: $\nabla f(x) + \sum \lambda_i \nabla g_i(x) + \sum \mu_j \nabla h_j(x) = 0$ .
Primal Feasibility: $g_i(x) \leq 0$ , $h_j(x) = 0$ .
Dual Feasibility: $\lambda_i \geq 0$ .
Complementary Slackness: $\lambda_i g_i(x) = 0$ .

Key Equations:

Gradient Descent Update:

x_{k+1} = x_k - \eta \nabla f(x_k)

KKT Lagrangian:

L(x, \lambda, \mu) = f(x) + \sum \lambda_i g_i(x) + \sum \mu_j h_j(x)

Convexity Condition (Hessian):

\nabla^2 f(x) \succeq 0

5. Function Analysis (Convexity, Extrema)

Key Concepts:

Convex/Concave Functions:
- $f$ convex $\Rightarrow$ any local minimum is global.
- $f$ concave $\Rightarrow$ any local maximum is global.
Critical Points:
- Find where $\nabla f(x) = 0$ .
- Classify using second derivative test (Hessian).
Global Extrema:
- For $f(x) = x^4 - 4x^2 + 5$ :
  - Find critical points: $f'(x) = 4x^3 - 8x = 0$ .
  - Evaluate $f''(x)$ to determine minima/maxima.

Key Equations:

Second Derivative Test:
- If $\nabla^2 f(x) \succ 0$ : local minimum.
- If $\nabla^2 f(x) \prec 0$ : local maximum.
Example Function:

f(x) = x^4 - 4x^2 + 5 \implies f'(x) = 4x^3 - 8x

Critical points: $x = 0, \pm \sqrt{2}$ .

IIT M DIPLOMA FN EXAM QDD2 13 Apr 2025 - 13 Apr 2025

Answer key

Concepts this paper tests

Core Topics & Key Concepts

1. Probability & Statistics

Key Concepts:

Key Equations:

2. Linear Algebra & Matrix Decompositions

Key Concepts:

Key Equations:

3. Principal Component Analysis (PCA)

Key Concepts:

Key Equations:

4. Convex Optimization

Key Concepts:

Key Equations:

5. Function Analysis (Convexity, Extrema)

Key Concepts:

Key Equations:

2. Linear Algebra & SVD

3. PCA Workflow

4. Convex Optimization

5. Function Analysis