IIT M IMPROVEMENT AN EXAM QIB4 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 for the Diploma-Level Machine Learning Foundations (Computer-Based Exam), along with Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Concepts

A. Machine Learning Fundamentals

1. Supervised Learning

   • Classification (e.g., loan approval based on features like income, credit score).

     • Algorithms: Logistic Regression, Decision Trees, SVM.
     • Loss functions: Cross-entropy, Hinge loss.

   • Regression (e.g., polynomial regression for quadratic models).

     • Algorithms: Linear Regression, Polynomial Regression, Ridge/Lasso.

     • Loss functions: Mean Squared Error (MSE).

       • Equation:

• Least Squares Method for parameter estimation (θ₀, θ₁, θ₂ in $y = θ₀ + θ₁x + θ₂x²$).

  • Closed-form solution (Normal Equation):

• Model Evaluation:

  • Squared loss, reconstruction loss (e.g., autoencoders).
  • Overfitting/underfitting.

1. Unsupervised Learning

   • Clustering (e.g., K-Means, Hierarchical Clustering).

   • Dimensionality Reduction (e.g., PCA, Autoencoders).

     • Autoencoder loss (reconstruction loss):

where $f$ = encoder, $g$ = decoder.

1. Key Distinctions:

   • Classification vs. Regression vs. Clustering vs. Dimensionality Reduction.

B. Linear Algebra

1. Vectors

   • Orthogonality: Two vectors $v_1$ and $v_2$ are orthogonal if $v_1^T v_2 = 0$.
   • Linear Independence: Vectors are linearly independent if no vector can be written as a linear combination of others.
   • Projection: Projection of $v_1$ onto $v_2$:

• Parallel/Collinear Vectors: Vectors are parallel if one is a scalar multiple of the other.

1. Matrices

   • Eigenvalues/Eigenvectors:

     • $A v = \lambda v$, where $\lambda$ = eigenvalue, $v$ = eigenvector.

     • Properties:

       • Sum of columns = zero vector ⇒ $\lambda = 0$ is an eigenvalue.
       • Real symmetric matrices have real eigenvalues.
       • Orthogonal matrices: $A^T A = I$ (eigenvalues have magnitude 1).

     • Diagonalizability:

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

   • Determinant:

     • $\det(A)$ for a matrix where all non-zero vectors are eigenvectors with eigenvalue $\lambda$: $\det(A) = \lambda^n$.

   • Row Space/Basis:

     • Row space of $A$: Span of its rows.
     • Basis: Linearly independent rows that span the row space.

   • Linear Relations:

     • Example: $R_2 = 2R_1 + R_3$ (from matrix rows).

1. Special Matrices

   • Orthogonal Matrices: $A^T A = I$.

     • Example: Rotation matrix:

• Eigenvalues: $e^{i\theta}, e^{-i\theta}$ (complex if $\theta \neq 0, \pi$).

  • Symmetric Matrices: $A = A^T$.

C. Calculus & Optimization

1. Linear Approximation

   • For $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, the linear approximation at $(a, b)$:

• Example for $f(x_1, x_2) = e^{x_1} + x_2$ at $(0,0)$:

• Plane Equation: $w^T x + b = 0$, where $w$ is the normal vector.

1. Gradient Descent

   • Update rule for regression:

D. Geometry

1. Contour Plots

   • Visualizing level sets of $f(x_1, x_2)$.
   • Example: Contours of $f(x_1, x_2) = e^{x_1} + x_2$ are horizontal shifts of exponential curves.

2. Key Equations

Graph 2: Linear Algebra Concepts

Graph 3: Calculus & Optimization

Graph 4: Relationships Between Topics

4. Exam-Specific Tips

1. Polynomial Regression:

   • For $y = \theta_0 + \theta_1 x + \theta_2 x^2$, construct the design matrix $X$ with columns $[1, x, x^2]$ and solve $\theta = (X^T X)^{-1} X^T y$.

1. Eigenvalues/Eigenvectors:

   • If all non-zero vectors are eigenvectors with eigenvalue $\lambda$, the matrix is $A = \lambda I$, so $\det(A) = \lambda^n$.

1. Linear Approximation:

   • For $f(x_1, x_2) = e^{x_1} + x_2$, compute gradients at $(0,0)$:

So, $L_{(0,0)}[f](x) = f(0,0) + \nabla f \cdot x = 1 + x_1 + x_2$.

1. Reconstruction Loss:

   • Encode each data point, decode it, and compute the mean squared distance from the original.

1. Orthogonal Matrices:

   • Rotation matrices are orthogonal and have eigenvalues $e^{i\theta}$, which are complex unless $\theta = 0$ or $\pi$.

1. Basis for Row Space:

   • Perform row reduction (Gaussian elimination) to find linearly independent rows.