MFML Lecture to Course Content Map

MFML Lecture to Course Content Map #

This file maps the uploaded Maths lecture PDFs and webinar PDFs against the official course handout/contact-session plan. It is intended as an exam preparation index and as a source map for future Hugo Markdown notes.

Course identity #

• Course: Mathematical Foundations for Machine Learning
• Course code: AIML ZC416
• Main areas: linear algebra, vector spaces, matrix decompositions, vector calculus, optimisation, PCA, and SVM.

Official module structure #

Contact session by lecture #

Webinar mapping #

Mid-Sem focus #

The course handout states that the Mid-Semester Test covers Weeks 1-8. So for Mid-Sem, focus on:

MFML Exam Revision Index

MFML Exam Revision Index #

This is a practical revision index for the uploaded Mathematical Foundations for Machine Learning material.

Exam split #

High-priority concept checklist #

Linear systems and matrices #

• Convert equations into matrix form Ax = b
• Understand solution types: no solution, unique solution, infinite solutions
• Identify pivot and free variables
• Understand row operations, REF/RREF, rank, nullity
• Know matrix inverse and transpose properties

Vector spaces #

• Definition of vector space and subspace
• Closure under addition and scalar multiplication
• Span, linear combination, linear independence
• Basis, dimension, rank
• Column space, row space, nullspace

Analytic geometry #

• Norm properties
• Manhattan norm and Euclidean norm
• Inner product definition
• Symmetric positive-definite matrices
• Distance, angle, orthogonality
• Orthonormal basis and Gram-Schmidt

Matrix decompositions #

• Determinant and trace
• Cofactor expansion
• Row operation effect on determinant
• Eigenvalue equation Av = λv
• Characteristic equation det(A - λI) = 0
• Diagonalisation A = PDP^{-1}
• Spectral theorem for symmetric matrices
• Cholesky decomposition
• SVD A = UΣV^T
• Low-rank approximation

Vector calculus #

• Derivative from first principles
• Partial derivatives
• Gradient as direction of steepest ascent
• Gradient of vector-valued functions
• Matrix-gradient identities
• Chain rule
• Backpropagation and automatic differentiation

Taylor series and Hessian #

• Taylor polynomial
• Taylor series and Maclaurin series
• Remainder term
• Taylor series in two variables
• Hessian matrix
• First derivative and second derivative tests
• Maxima, minima and saddle points

Gradient descent and optimisation #

• Negative gradient direction
• Learning rate/step size
• Line search
• Convergence and local minima
• Constrained vs unconstrained optimisation
• Lagrange multipliers
• Convex optimisation
• SGD and optimisation in ML
• Feature preprocessing and scaling
• Overfitting in optimisation examples

Nonlinear optimisation algorithms #

• Difficult surfaces: cliffs, valleys, flat regions
• Curvature and why first-order methods can struggle
• Momentum update and intuition
• AdaGrad
• RMSProp
• Adam
• Learning rate decay

PCA #

• Dimensionality reduction problem
• Centred data and covariance matrix
• Maximum variance view
• Projection/reconstruction view
• Principal components as eigenvectors of covariance matrix
• SVD relation to PCA
• Low-rank approximation and Eckart-Young theorem
• PCA in high dimensions
• Practical PCA steps

SVM #

• Linear classifiers
• Margin and support vectors
• Hard-margin SVM primal formulation
• Lagrangian for SVM
• KKT conditions
• Primal vs dual perspective
• Role of inner products
• Kernel trick
• Hinge loss
• Soft-margin SVM

Suggested revision order #

Phase 1: Foundations #

1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Webinar 1 problems related to REF, nullspace, column space and subspaces

Phase 2: Matrix decompositions #

1. Lecture 4
2. Lecture 5
3. Webinar 1 and Webinar 2 eigenvalue/eigendecomposition problems

Phase 3: Calculus and optimisation foundations #

1. Lecture 6
2. Lecture 7
3. Lecture 8
4. Webinar 2 maxima/minima and Hessian problems

Phase 4: Optimisation for ML #

1. Lecture 9
2. Lecture 10
3. Lecture 11
4. Webinar 3 gradient-descent step-size problems

Phase 5: PCA and SVM #

1. Lecture 12
2. Lecture 13
3. Lecture 14
4. Lecture 15
5. Webinar 4 / SVM problems

What to ask me next #

Use these prompts when generating Hugo pages:

MFML Topic to Source Index

MFML Topic to Source Index #

This index tells you where to look when you want to create future notes or revise a topic.

Primal and Dual Perspective for Linear SVM

May 28, 2026

Primal and Dual Perspective for Linear SVM #

A linear Support Vector Machine finds a hyperplane that separates two classes with the maximum possible margin.

The primal view gives the direct geometric optimisation problem. The dual view rewrites the problem using Lagrange multipliers and reveals why only support vectors matter.

Key takeaway: Linear SVM maximises the margin by minimising

\( \frac{1}{2}\|w\|^2 \)

subject to correct-classification constraints. The dual solution expresses ( w )

Mathematical Preliminaries for SVM

May 28, 2026

Mathematical Preliminaries for SVM #

Support Vector Machines use optimisation, geometry and kernels. Before deriving SVM, we need constrained optimisation, Lagrange multipliers, primal and dual problems, KKT conditions, hyperplanes and kernel functions.

Key takeaway: SVM is built on constrained optimisation. The hard-margin SVM primal problem is a quadratic optimisation problem with linear inequality constraints. The dual problem uses Lagrange multipliers and leads naturally to support vectors and kernels.

Nonlinear SVM

May 28, 2026

Nonlinear SVM #

A linear SVM works well when the data can be separated by a straight line or hyperplane. When the data is not linearly separable in the original input space, nonlinear SVM maps the data to a higher-dimensional feature space where a linear separator may exist.

Key takeaway: Nonlinear SVM uses the kernel trick. Instead of explicitly mapping

\( x \)

to ( \phi(x) )