70112

Non-Euclidean Methods in Machine Learning

Module aims

We understand the world by interaction with the bodies we observe. This Kantian empirical realism called experience is made possible by the a priori Euclidean constraints on space. While being subject to limits of scales and tolerances of our senses, such a flat view of the world has been the driving force in many fields laying the foundations of the first AI systems. Recent progress is challenging this perspective by showing that the world around us as well as the answers we are looking for admit a non-Euclidean, curved structure. Hence, it becomes desirable for our machine learning models to adapt naturally. In this course, we cover the nuts and bolts of learning non- Euclidean embeddings connecting non-Euclidean domains and parameter spaces. We will apply optimization techniques from Riemannian geometry, bring in knowledge from graph theory and present novel developments in neural networks that are suited to data living on lower dimensional manifolds.

Learning outcomes

Upon successful completion of this module you will be able to:
- Evaluate geometric machine learning as a tool to model common learning frameworks.
- Design optimizers on Riemannian manifolds to implement smooth constrained optimization.
- Synthesise discrete operators on graphs from their continuous versions.
- Modify learning models to operate on constrained domains and outcomes.
- Implement deep learning on unstructured domains such as graphs, point sets and meshes.
- Implement mechanisms to yield structured output from learning models.

Module syllabus

Review of tensors, differential and Riemannian geometry.
Optimization on manifolds
Probability and Bayesian inference on Manifolds
Going from manifolds to graphs: continuous to discrete

Teaching methods

The material will be taught through traditional lectures, backed up by group projects, coding and theoretical tasks, with an algorithmic perspective. You will receive technical support for the various coding tasks from Graduate Teaching Assistants (GTAs). An online discussion forum will be used for the module.

Assessments

There will be a number of coursework exercises contributing a total of 30% and an exam contributing to 70% of the marks. These will assess both theoretical and practical aspects of the subject.               
Written and verbal feedback will be provided throughout the module. Detailed written feedback will be provided on each coursework. Class-wide exam feedback will be provided after the exam.    

Module leaders