Deep learning identifies topological equivalence of group actions on surfaces
Deep Learning of Topological Equivalence Classes
Machine Learning
Summary
This work shows how deep learning can help tell when two mathematical descriptions of shapes called Riemann surfaces are essentially the same from a topological point of view. The authors use neural network techniques originally made for understanding language and images to solve a pure math problem involving finite group actions. They also explain how the structure of the mathematical objects influenced their design of the deep learning model. Lastly, they analyze what features the model learned about these topological equivalence classes.
What this means in practice
- •For mathematical software developers: Create software that uses deep learning to classify group actions on surfaces to aid mathematical research and automated theorem verification.
- •For machine learning engineers: Design specialized neural network architectures that leverage algebraic structures for improving model interpretability and performance on abstract problems.
Authors
Linden Disney-Hogg, Max Zuev
Abstract
We demonstrate an application of deep learning to the problem of identifying whether two generating vectors correspond to topologically equivalent finite group actions on Riemann surfaces, showing how architectures developed for semantic understanding have use for pure mathematical problems. In doing so we discuss how desired algebraic structure influences model design, and explore which structural properties of topological equivalence classes the model has successfully learnt.