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 #

Module	Course handout area	Main ideas	Uploaded lecture coverage
M1	Solution of linear systems	Systems of equations, matrices, solving Ax = b	Lecture 1, Webinar 1
M2	Vector spaces and analytic geometry	Vector spaces, linear independence, basis, rank, norms, inner products, angles, orthogonality, orthonormal basis	Lecture 2, Lecture 3, Webinar 1
M3	Matrix decomposition methods	Determinant, trace, eigenvalues, eigenvectors, Cholesky, eigendecomposition, diagonalisation, SVD, matrix approximation	Lecture 4, Lecture 5, Webinar 1, Webinar 2
M4	Vector calculus	Univariate differentiation, partial derivatives, gradients, matrix gradients, Taylor/Maclaurin series, Hessian, backpropagation, automatic differentiation	Lecture 6, Lecture 7, Lecture 8, Webinar 2
M5	Continuous optimisation	Gradient descent, constrained optimisation, Lagrange multipliers, convex optimisation	Lecture 9, Lecture 14, Webinar 2, Webinar 3, Webinar 4
M6	Nonlinear optimisation	Learning rate, initialisation, SGD, feature preprocessing, local optima, cliffs/valleys, momentum, AdaGrad, RMSProp, Adam	Lecture 10, Lecture 11, Webinar 3
M7	Dimensionality reduction, PCA, SVM	PCA perspectives, low-rank approximation, high-dimensional PCA, practical PCA, SVM preliminaries, primal/dual SVM, kernels	Lecture 12, Lecture 13, Lecture 14, Lecture 15, Webinar 4

Contact session by lecture #

Session	Course handout topic	Uploaded file	What the lecture appears to cover	Exam relevance
1	Solution of linear systems	`Lecture_1.pdf`	Linear algebra introduction, closure, systems of linear equations, matrix representation, solution types: no solution, unique solution, infinite solutions, pivot/free variables, matrix operations, inverse, transpose, compact Ax=b form	Very high for Mid-Sem and Comprehensive
2	Vector spaces, linear independence, basis, rank	`Lecture_2.pdf`	Groups, Abelian groups, vector spaces, vector subspaces, closure tests, linear combinations, span, linear independence, basis, rank, nullspace/column space ideas	Very high for Mid-Sem and Comprehensive
3	Analytic geometry	`Lecture_3.pdf`	Norms, dot product, inner products, bilinear mappings, symmetric positive-definite matrices, lengths, distances, angles, orthogonality, orthonormal basis, Gram-Schmidt ideas	Very high for Mid-Sem and Comprehensive
4	Matrix Decomposition I	`lecture_4.pdf`	Determinant, cofactor formula, determinant behaviour under row operations, rank-det relation, eigenvalues/eigenvectors, Cholesky-related positive definite ideas	Very high for Mid-Sem and Comprehensive
5	Matrix Decomposition II	`lecture_5.pdf`	Diagonal matrices, diagonalisation, eigendecomposition, spectral theorem for symmetric matrices, SVD, matrix approximation	Very high for Mid-Sem and Comprehensive
6	Vector Calculus I	`lecture_6.pdf`	Differentiation of univariate functions, polynomial derivatives, Taylor polynomial/series, partial derivatives, gradients, vector-valued gradients	Very high for Mid-Sem and Comprehensive
7	Vector Calculus II	`lecture_7_edited.pdf`	Matrix gradients, useful gradient identities, backpropagation, automatic differentiation, chain rule through neural-network layers	High for Mid-Sem and Comprehensive
8	Vector Calculus III	`lecture_8.pdf`	Taylor/Maclaurin series theory, remainder term, two-variable Taylor series, Hessian matrix, maxima/minima, unconstrained optimisation preliminaries	Very high for Mid-Sem and Comprehensive
9	Continuous Optimisation	`Lecture_9.pdf`	Gradient descent, negative gradient direction, local minima, step size, line search, convergence intuition, quadratic examples	Very high for Comprehensive; likely useful for quizzes/problems
10	Nonlinear Optimisation I	`Lecture_10.pdf`	Initialisation, objective functions in ML, overfitting, feature processing/preprocessing, SGD and practical optimisation behaviour	High for Comprehensive
11	Nonlinear Optimisation II	`Lecture_11.pdf`	Difficult topologies: cliffs, valleys, flat regions, curvature; momentum, AdaGrad, RMSProp, Adam	High for Comprehensive
12	PCA I	`Lecture_12.pdf`	Dimensionality reduction, PCA problem setting, centred data, covariance, maximum variance perspective, projection perspective	Very high for Comprehensive
13	PCA II	`Lecture_13.pdf`	Practical PCA, eigenvector computation, SVD relationship, low-rank approximation, high-dimensional PCA, key PCA steps	Very high for Comprehensive
14	Mathematical preliminaries for SVM	`Lecture 14.pdf`	Constrained optimisation, Lagrangian, quadratic programming, primal/dual, weak/strong duality, Slater condition, KKT conditions, kernels, linear classifiers	Very high for Comprehensive
15	Primal/dual linear SVM	`Lecture_15.pdf`	SVM primal problem, dual formulation, KKT conditions, support vectors, hinge loss, linear SVM numerical problem, hard/soft-margin direction	Very high for Comprehensive
16	Nonlinear SVM / kernels	Not clearly uploaded as a separate Lecture 16 PDF	Kernel functions, nonlinear SVM examples; likely partly covered in Lecture 14/15 and webinars	Very high for Comprehensive; gap to fill if Lecture 16 exists

Webinar mapping #

Webinar file	Main role	Best linked lectures	Exam use
`Webinar_1.pdf`	Problem sheet on linear systems, REF/RREF, column space, nullspace, row independence, subspaces, inner products, Cauchy-Schwarz, Cholesky, eigenvalues	Lectures 1-5	Excellent for Mid-Sem problem practice
`Webinar_2.pdf`	Worked problems on maxima/minima, eigenvalues/spectral decomposition, gradient-related calculations and PCA-style examples	Lectures 4-9, 12-13	Excellent for Mid-Sem revision and Comprehensive practice
`Webinar_3.pdf`	Gradient descent algorithm, step-size derivation for quadratic functions, worked gradient descent examples	Lectures 8-11	Excellent for optimisation exam problems
`webinar_4.pdf`	Appears linked to optimisation/SVM/PCA practice based on uploaded set; use as problem-solving supplement after Lecture 12 onwards	Lectures 12-15	Comprehensive exam practice

Mid-Sem focus #

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

Lecture 1: Linear systems and matrices
Lecture 2: Vector spaces, subspaces, linear independence, basis, rank
Lecture 3: Norms, inner products, orthogonality, Gram-Schmidt
Lecture 4: Determinant, trace, eigenvalues, eigenvectors, Cholesky
Lecture 5: Diagonalisation, eigendecomposition, SVD, matrix approximation
Lecture 6: Differentiation, partial derivatives, gradients, Taylor series
Lecture 7: Matrix gradients, gradient identities, backpropagation, automatic differentiation
Lecture 8: Taylor theorem, Hessian, maxima/minima, unconstrained optimisation
Webinar 1 and Webinar 2 for worked problem practice

Comprehensive exam focus #

The course handout states that the Comprehensive Exam covers all sessions 1 to 16. So for Comprehensive, add:

Lecture 9: Gradient descent and step size
Lecture 10: Nonlinear optimisation, initialisation, feature preprocessing, overfitting, SGD
Lecture 11: Cliffs, valleys, momentum, AdaGrad, RMSProp, Adam
Lecture 12: PCA I
Lecture 13: PCA II
Lecture 14: SVM mathematical preliminaries, Lagrangian, KKT, duality
Lecture 15: Linear SVM, support vectors, hinge loss, primal/dual solution
Missing/gap area: nonlinear SVM and kernels if a separate Lecture 16 exists

Suggested future Hugo Markdown pages #

These are good page boundaries for your website and revision notes:

01-linear-systems-and-matrices.md
02-vector-spaces-subspaces-basis-rank.md
03-analytic-geometry-norms-inner-products.md
04-determinants-trace-eigenvalues.md
05-eigendecomposition-svd-matrix-approximation.md
06-vector-calculus-gradients.md
07-backpropagation-automatic-differentiation.md
08-taylor-series-hessian-maxima-minima.md
09-gradient-descent-continuous-optimisation.md
10-nonlinear-optimisation-sgd-feature-preprocessing.md
11-momentum-adagrad-rmsprop-adam.md
12-pca-foundations.md
13-pca-practical-computation-svd.md
14-lagrangian-duality-kkt.md
15-support-vector-machines.md
16-nonlinear-svm-kernels.md
99-mfml-exam-formula-sheet.md
99-mfml-webinar-problem-bank.md

Important exam-preparation note #

For exam preparation, do not only read the lecture slides. Use this sequence:

Read the lecture slides for concept flow.
Create a one-page formula sheet for each lecture.
Work through the matching webinar problems.
Mark every problem by topic: definition, proof, calculation, interpretation, or algorithm.
For open-book comprehensive exam, prepare indexed printed notes rather than loose sheets.