IIT M IMPROVEMENT AN EXAM QIA3 26 Oct - 26 Oct 2025

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. Machine Learning Foundations

• Supervised Learning:

  • Classification (e.g., loan approval based on features like income, credit score). Key Models: Logistic Regression, Decision Trees, SVM.
  • Regression (e.g., polynomial regression for quadratic models). Key Equation:

Least Squares Method: Minimize $\sum (y_i - \hat{y}_i)^2$.

• Loss Functions:

  • Squared Loss: $L = \sum (y_i - f(x_i))^2$.
  • Reconstruction Loss (Autoencoders): Mean squared distance between input and reconstructed input.

• Unsupervised Learning:

  • Clustering (e.g., K-Means, Hierarchical Clustering).
  • Dimensionality Reduction (e.g., PCA, Autoencoders).

• Model Evaluation:

  • Overfitting/underfitting, bias-variance tradeoff.
  • Linear separability (e.g., whether a linear model can achieve zero squared loss).

2. Linear Algebra

• Vectors:

  • Orthogonality: Two vectors $v_1, v_2$ are orthogonal if $v_1^T v_2 = 0$.
  • Projection of $v_1$ onto $v_2$:

• Parallel vectors: Scalar multiples of each other ($v_1 = k v_2$).

• Matrices:

  • Eigenvalues/Eigenvectors:

    • Definition: $A v = \lambda v$.
    • Key Insight: If the sum of columns of $A$ is the zero vector, $\lambda = 0$ is an eigenvalue.
    • Real symmetric matrices have real eigenvalues.

  • Diagonalizability:

    • A matrix is diagonalizable if it has $n$ linearly independent eigenvectors (not necessarily distinct eigenvalues).
    • False Statement: "A matrix is diagonalizable iff it has $n$ distinct eigenvalues."

  • Orthogonal Matrices:

    • Definition: $A^T A = I$.
    • Example: Rotation matrix $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$.
    • Properties: Preserves norms, eigenvalues lie on the unit circle (no real eigenvalues for $0 < \theta < 180^\circ$).

  • Determinant:

    • For a matrix where every non-zero vector is an eigenvector with eigenvalue $\lambda$, $A = \lambda I$.
    • $\det(A) = \lambda^n$ (for $n \times n$ matrix).

• Row Space & Basis:

  • Row space of $A$: Span of its rows.
  • Basis: Linearly independent rows that span the row space.
  • Example: For $A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & 1 & 5 \\ 2 & -1 & 1 \end{bmatrix}$, a basis is $\{(1,0,2), (0,1,-1), (0,0,-1)\}$.

• Linear Relations:

  • Example: $R_2 = 2R_1 + R_3$ for rows of $A$.

3. Calculus & Optimization

• Linear Approximation:

  • For $f(x_1, x_2) = e^{x_1} + x_2$, the linear approximation at $(0,0)$ is:

• Contours of $f$: Curves where $f(x_1, x_2) = c$ (exponential decay in $x_1$, linear in $x_2$).

• Plane Equation:

  • For a plane $w^T x + b = 0$ with $\|w\| = \sqrt{3}$, if the plane represents $L_{(0,0)}[f](x) = 1 + x_1 + x_2$, then $w = (1,1,-1)$ and $b = -1$ (but normalized to $w^T w = 3$ gives $b = 1$).

4. Autoencoders (Dimensionality Reduction)

• Encoder-Decoder:

  • Encoder: $f(x_1, x_2, x_3) = 0.5x_1 + 0.5x_2 - x_3$.
  • Decoder: $g(u) = [u, u, u]$.
  • Reconstruction loss: Mean squared distance between original and reconstructed inputs. Example Calculation:

Key Equations Summary

2. Linear Algebra Concepts

3. Calculus & Optimization

4. Autoencoders