Non-Archimedean Polydisc Spaces and Applications to Optimisation

2026-06-05

AI Summary

This work addresses the lack of geometric structures in non-Archimedean spaces suitable for hierarchical data optimization. We propose a novel framework based on Berkovich geometry—specifically, a non-Archimedean polydisk space constructed as a product of closed balls—which naturally supports hierarchical representations and is introduced here for the first time into optimization. We prove that this space admits unique geodesics and can be isometrically embedded into a metric tree. A class of objective functions with piecewise polynomial structure and universal approximation capability is defined, for which we establish existence theory of solutions and design corresponding optimization algorithms. Leveraging tools from non-Archimedean analysis, metric geometry, and polynomial absolute value functions, we implement an open-source Julia library and demonstrate through experiments the effectiveness of our approach for hierarchical data optimization.

Abstract

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.

Problem

• non-Archimedean
• optimisation
• polydisc spaces
• metric geometry
• hierarchical data

Innovation

• non-Archimedean optimization
• polydisc spaces
• Berkovich geometry
• metric trees
• piecewise polynomial functions