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This paper introduces a novel mathematical framework for optimization over non-Archimedean spaces, leveraging Berkovich geometry to handle hierarchical data structures like metric trees. By proving the existence of minimizers and providing a universal approximation property for specific function classes, the authors establish a theoretical foundation compatible with classical optimization techniques. The release of an open-source Julia library allows practitioners to immediately test these algorithms for complex hierarchical problems.
Non-Archimedean Polydisc Spaces and Applications to Optimisation
arXiv:2606.07782v1 Announce Type: cross 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.
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.
54 pages, 23 figures. Comments welcome
Optimization and Control (math.OC); Machine Learning (cs.LG); Metric Geometry (math.MG)
arXiv:2606.07782 [math.OC]
(or arXiv:2606.07782v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2606.07782 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 259 words