Non-Archimedean Polydisc Spaces and Applications to Optimisation
For researchers in optimisation and non-Archimedean geometry, this work proposes a novel geometric structure for hierarchical data optimisation, but the practical impact is unclear as no empirical results or comparisons are provided.
The paper introduces polydisc spaces over non-Archimedean fields as a new framework for optimisation, showing they embed metric trees and admit piecewise polynomial functions with universal approximation. It proves existence of minimisers and provides algorithms with an open-source Julia library.
We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.