IIT M FOUNDATION AN3 EXAM QPF3 16 JULY 2023 - 16 Jul 2023

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Summary of Core Topics, Concepts, Principles, and Equations

1. Linear Algebra

• Matrix Operations and Properties

  • Row space, column space, null space, and their orthogonality.
  • Eigenvectors and eigenvalues.
  • Diagonalizable matrices and their properties.
  • Projection matrices and their determinants.

• Key Equations

  • $A \in \mathbb{R}^{d \times d}$ is diagonalizable.
  • $X_{n+1} = X_n - h \nabla f(X_n)$ (Gradient Descent Algorithm).
  • $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$ (Gradient of a function).
  • $\nabla f \cdot \hat{n}$ (Directional Derivative).

2. Machine Learning

• Linear Regression

  • Least squares method.
  • Loss functions: Mean Squared Error (MSE), Maximum Absolute Error.
  • Projection of vectors and least squares solutions.

• Dimensionality Reduction

  • Encoder and decoder functions.
  • Reconstruction error: $R(f, g) = \frac{1}{n} \sum_{i=1}^{n} \| x_i - g(f(x_i)) \|^2$.

• Key Equations

  • $f(X) = w^T x$ (Linear Regression Model).
  • $L = \max_i |f(x^i) - y^i|$ (Maximum Absolute Error Loss).
  • $R(f, g) = \frac{1}{n} \sum_{i=1}^{n} \| x_i - g(f(x_i)) \|^2$ (Reconstruction Error).

3. Calculus

• Directional Derivatives

  • Gradient and its evaluation.
  • Dot product of gradient and direction vector.

• Key Equations

  • $\nabla f \cdot \hat{n}$ (Directional Derivative).
  • $f(x, y) = ye^{xy}$ (Linear Approximation).

4. Optimization

• Gradient Descent

  • Algorithm for minimizing functions.
  • Step size and convergence.

• Key Equations

  • $X_{n+1} = X_n - h \nabla f(X_n)$ (Gradient Descent Update Rule).