Bachelor of Science (BSc)

BSc Computer Science and Mathematics

One of the most sought-after subject combinations in industry, this course is designed to provide the perfect balance of creativity and logic.
  • Duration: 3 years
  • Year of entry: 2025
  • UCAS course code: GG14 / Institution code: M20
  • Key features:
  • Scholarships available

Full entry requirementsHow to apply

Fees and funding

Fees

Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £36,000 per annum. For general information please see the undergraduate finance pages.

Policy on additional costs

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

Scholarships/sponsorships

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our  undergraduate student finance pages .

Course unit details:
Mathematical Topics in Machine Learning

Course unit fact file
Unit code COMP34312
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 2
Available as a free choice unit? Yes

Overview

Topic 1: Empirical risk minimization, regularisation; bias/variance theory and the relation to overfitting; probabilistic view: likelihood vs loss, introducing exponential families.

 

Topic 2: Information theory: KL-divergence vs. cross-entropy, mutual information; the view of ML as compression.

 

Topic 3: Optimization theory (calculus). Why GD?  What are convex and non-convex functions? How do gradients inform how we optimize a function? How can we use second order properties? How can we prove whether a method will converge?

 

Topic 4: Dimensionality reduction (matrix algebra). “refine, denoise, and visualise your data”.  Data usually has limited degrees of interest, living on a low-dimensional manifold within a high-dimensional space. This topic will introduce students to matrix-algebra-intensive methods used to learn feature dimensions that can aid in your model fitting process. Examples include PCA, spectral embedding, Fisher discriminant analysis, etc. These allow visualisation, denoising and enhancing the separation of data for classification.

Pre/co-requisites

Unit title Unit code Requirement type Description
Machine Learning COMP24112 Pre-Requisite Compulsory
COMP34312 has a pre-requisite of COMP24112

To enrol students are required to have taken COMP24112

Aims

Machine Learning has certain mathematical “building blocks”, which turn up in the study of all types of models and algorithms.  Specifically, these building blocks utilise techniques from probability theory, matrix algebra, and calculus.

 

This module aims to introduce students to these, and then show how to: (1) read and correctly interpret research papers in this context; and (2) understand how novel algorithms are devised in modern ML.

 

There will be no required coding/practical algorithm development.  The module aims to be a stepping-stone toward research, either in industry or in a PhD.

Learning outcomes

  • Discuss key mathematical terms in ML, e.g. bias/variance, entropy/cross-entropy, regularisation, the duality between the probabilistic vs. loss function view of ML, and their consequences in practical scenarios
  • Correctly manipulate and interpret mathematical expressions for the likelihood of models, entropies and mutual information between random variables.
  • Explain taught linear algebra concepts and methods, e.g., vector space/subspace, basis, linear independence, rank, inverse, orthogonality, singular value decomposition, eigen-decomposition.
  • Explain and compare the nature and advantages/disadvantages of dimensionality reduction methods, e.g., PCA, spectral embedding, FDA, and how they make use of linear algebra concepts.
  • Discuss and interpret data / concepts on convex and non-convex optimisation, including convergence properties and proof techniques to explain stochastic gradient descent.

Syllabus

Topic 1: Empirical risk minimization, regularisation; bias/variance theory and the relation to overfitting; probabilistic view: likelihood vs loss, introducing exponential families.

 

Topic 2: Information theory: KL-divergence vs. cross-entropy, mutual information; the view of ML as compression.

 

Topic 3: Optimization theory (calculus). Why GD?  What are convex and non-convex functions? How do gradients inform how we optimize a function? How can we use second order properties? How can we prove whether a method will converge?

 

Topic 4: Dimensionality reduction (matrix algebra). “refine, denoise, and visualise your data”.  Data usually has limited degrees of interest, living on a low-dimensional manifold within a high-dimensional space. This topic will introduce students to matrix-algebra-intensive methods used to learn feature dimensions that can aid in your model fitting process. Examples include PCA, spectral embedding, Fisher discriminant analysis, etc. These allow visualisation, denoising and enhancing the separation of data for classification.

Teaching and learning methods

Unit will consist of 4 major topics, delivered in 2-week blocks before Easter.  Each topic will consist of videos to watch, and readings to cover, before the interactive sessions. The interactive sessions will act to reinforce the videos/readings.  Weekly MCQs in class will be used as formative and summative assessments.

 

After Easter, a series of carefully selected classic research papers will be read week by week in groups, introducing students to the methods in how to read/interpret research results.  Presentations of the papers will consolidate the depth of understanding.

Assessment methods

Method Weight
Written exam 80%
Written assignment (inc essay) 20%

Feedback methods

Correct answers discussed the following week

Recommended reading

Selected chapters: Machine Learning, A Probabilistic Perspective by Kevin Murphy;

Selected chapters: Probability in Data Science, by Sidney Chan

Study hours

Scheduled activity hours
Assessment written exam 2
Lectures 11
Practical classes & workshops 11
Independent study hours
Independent study 76

Teaching staff

Staff member Role
Gavin Brown Unit coordinator

Return to course details