Sheard, David;
(2022)
Nielsen equivalence in Coxeter groups; and a geometric approach to group equivariant machine learning.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
In this thesis we study two main threads. In Part I, we initiate the study of Nielsen equivalence in Coxeter groups—the classification of finite generating sets up to a natural action of the automorphism group of a free group. We explore different Nielsen equivalence invariants and adapt the method of Lustig and Moriah [79] to the Coxeter case. We also adapt the completion sequences of Dani and Levcovitz [31] to give a method of testing when generating sets of right-angled Coxeter groups are Nielsen equivalent. Coxeter systems have a distinguished set of elements, called the reflections, from which generating sets can be drawn. We study generating sets of reflections separately. In this case, the natural notion of equivalence is generated by partial conjugations of one generator by another. This arises naturally for Weyl groups in the context of cluster algebras via quiver mutations [6]. We study this mutation equivalence for Weyl groups, and reflection equivalence for arbitrary Coxeter systems. In the latter case we leverage hyperplane arrangements in the Davis complex associated to a Coxeter system to give geometric criteria from when a set of reflections generates and a test for when generating sets of reflections are reflection equivalent. In Part II, we discuss the other main topic of the thesis is group equivariant machine learning, based on joint work with Aslan and Platt [3]. We propose a novel approach to defining machine learning algorithms for problems which are equivariant with respect to some discrete group action. Our approach involves pre-processing the input data from a learning algorithm by projecting it onto a fundamental domain for the group action. We give explicit and efficient algorithms for computing this projection. We test our approach on three example learning problems, and demonstrate improvements in accuracy over other methods in the literature.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Nielsen equivalence in Coxeter groups; and a geometric approach to group equivariant machine learning |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10162247 |
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