IIT M DIPLOMA ET1 EXAM QPD1 S2 30 Apr 2023 - 30 Apr 2023

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.

1. Machine Learning Foundations

Core Topics:

• Supervised Learning:

  • Classification vs. Regression:

    • Classification: Maps input $x$ to discrete output $y$ (e.g., decision boundaries).
    • Regression: Maps input $x$ to continuous output $y$ (e.g., best-fit line).

    • Key Equation:

      • Regression: $y = w^T x + b$ (linear regression).
      • Classification: $\hat{y} = \text{sign}(w^T x + b)$ (linear classifier).

  • Types of Classifiers:

    • Linear (e.g., SVM, logistic regression) vs. Non-linear (e.g., kernel SVM, decision trees).

  • Types of Regression:

    • Linear, polynomial, ridge/lasso (regularized).

• Unsupervised Learning:

  • PCA (Principal Component Analysis):

    • Projection onto lines/planes to minimize reconstruction error.

    • Key Equation:

      • Covariance matrix: $C = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)(x_i - \mu)^T$.
      • Reconstruction error = sum of discarded eigenvalues.
    • Example: For $C = \begin{bmatrix} 4 & 2 \\ 2 & 4 \end{bmatrix}$, error = $\lambda_2 = 2$.

• Optimization:

  • Linear Programming:

    • Minimize/maximize a linear objective subject to constraints.

    • Example:

      • Minimize $50x_1 + 80x_2$ s.t. $x_1 + x_2 > 12$, $x_1 \leq 6$, $x_2 \leq 10$.

  • Non-linear Optimization:

    • Utility maximization (e.g., $\text{maximize } xy$ s.t. $3x + 4y = 200$).

2. Mathematics for ML

Core Topics:

• Linear Algebra:

  • Matrix Rank:

    • Full rank ($\text{rank}(A) = \min(m, n)$) implies unique solutions for $Ax = b$ if $m \geq n$.
    • Example: $A$ (10×8, rank 8) has a unique solution for some $b \in \mathbb{R}^{10}$.

  • Eigenvalues/Eigenvectors:

    • $A$ invertible iff no eigenvalue is 0.
    • $\lambda$ of $A$ ⇒ $\lambda^n$ of $A^n$.
    • Hermitian matrices: Eigenvectors orthogonal; eigenvalues real.

    • Key Equations:

      • $Av = \lambda v$.
      • $A = Q \Lambda Q^T$ (spectral decomposition for symmetric $A$).

  • Positive Definite Matrices:

    • $A$ symmetric positive definite ⇒ $x^T A x > 0$ for all $x \neq 0$.
    • $Q^T A Q$ preserves eigenvalues and positive definiteness.

• Multivariable Calculus:

  • Linear Approximation:

    • $f(x) \approx f(a) + \nabla f(a)^T (x - a)$.
    • Example: For $f(x, y, z) = e^{(x+y+z)^2}$, gradient at $(0, 0, 1)$ is $[2e, 2e, 2e]$.
    • Approximation at $(1, 1, 1)$: $f(1,1,1) \approx e + [2e, 2e, 2e] \cdot [1, 1, 0] = 5e$.

  • Taylor Series:

    • First-order: $f(x) \approx f(a) + \nabla f(a)^T (x - a)$.
    • Example: $f([3,2,1]) \approx 10 + [2,1,3] \cdot [2,1,0] = 15$.

• Probability & Statistics:

  • Random Variables:

    • Variance: $\text{Var}(X) = E[X^2] - (E[X])^2$.
    • Covariance: $\text{Cov}(X, Y) = E[XY] - E[X]E[Y]$.

    • Key Equation:

      • $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)$.

    • Example: Given $E[X] = 2$, $E[Y] = 3$, $E[X^2] = 9$, $E[Y^2] = 11$, $E[XY] = 8$:

      • $\text{Var}(X + Y) = (9-4) + (11-9) + 2(8-6) = 10$.

  • Joint Distributions:

    • Marginal/conditional PDFs, expectations.
    • Example: $P(0 < X < 1, 0 < Y < 1) = \int_0^1 \int_0^1 \frac{x^3 + y^3}{16} \, dx \, dy = \frac{1}{8}$.

  • Normal Distribution:

    • $X \sim \mathcal{N}(\mu, \Sigma)$ ⇒ $Y = a^T X \sim \mathcal{N}(a^T \mu, a^T \Sigma a)$.
    • Example: $Y = 2X_1 + X_2 \sim \mathcal{N}(20, 16)$.

  • Chebyshev’s Inequality:

    • $P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$.
    • For binary $X$, $\text{Var}(X) \leq \frac{1}{4}$.
    • Example: To ensure $P(|\bar{X} - p| < 0.01) \geq 0.95$, $n \geq 50000$.

• Convexity:

  • Convex Functions:

    • $f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$.
    • Examples: $e^x$, $x^2$, $-\log(x)$.
    • Non-convex: $\log(1 + x)$ (concave), $-x^2 e^{-x}$.

3. Key Equations Summary

Graph 2: Linear Algebra

Graph 3: Probability & Statistics

Graph 4: Calculus & Optimization

Graph 5: Convexity