Non-archimedean optimisation
Organizers
Speaker
Yue Ren
Time
Tuesday, June 17, 2025 3:00 PM - 4:00 PM
Venue
A3-4-312
Online
Zoom 928 682 9093
(BIMSA)
Abstract
Hierarchical data is ubiquitous in applications, from phylogenetics and natural language processing to clustering, decision trees, and network analysis. This data naturally forms tree-like structures that are poorly suited to Euclidean spaces, where representing them often requires an unreasonably large number of dimensions. For instance, in Large Language Models, each token indicates another dimension of the context window.
Non-Archimedean fields, such as the p-adic numbers, provide a more natural ambient space for these structures. But while these fields have been studied in number theory and algebraic geometry for over a century, they are largely unexplored in optimisation and machine learning, due to the initial challenges in adapting standard analytic tools.
In this talk, I discuss an effort towards bridging this gap using ideas from non-archimedean and tropical geometry. Our long-term goal is to establish practical techniques for optimisation over non-archimedean spaces, and ultimately enable machine learning over non-archimedean fields. Using such techniques it may become possible to exploit unique properties to non-Archimedean spaces, such as the lack of error accumulation.
This project is ongoing work with Julio Quijas Aceves (Durham), Oliver Clarke (Durham), Jeff Giansiracusa (Durham), Paul Lezeau (Google Deepmind), Anthea Monod (Imperial). It is part of the Erlangen AI Hub (https://erlangenhub.ox.ac.uk).
Non-Archimedean fields, such as the p-adic numbers, provide a more natural ambient space for these structures. But while these fields have been studied in number theory and algebraic geometry for over a century, they are largely unexplored in optimisation and machine learning, due to the initial challenges in adapting standard analytic tools.
In this talk, I discuss an effort towards bridging this gap using ideas from non-archimedean and tropical geometry. Our long-term goal is to establish practical techniques for optimisation over non-archimedean spaces, and ultimately enable machine learning over non-archimedean fields. Using such techniques it may become possible to exploit unique properties to non-Archimedean spaces, such as the lack of error accumulation.
This project is ongoing work with Julio Quijas Aceves (Durham), Oliver Clarke (Durham), Jeff Giansiracusa (Durham), Paul Lezeau (Google Deepmind), Anthea Monod (Imperial). It is part of the Erlangen AI Hub (https://erlangenhub.ox.ac.uk).