用非阿基米德空间中的多圆盘结构，构建可优化的分层数据模型。
Non-Archimedean Polydisc Spaces and Applications to Optimisation
- 引入多圆盘空间，融合非阿基米德场的层级结构与良好几何性质。
- 证明极小值存在且沿测地线函数为分段多项式，支持通用逼近。
- 适合研究分层数据优化的学者，开源库已支持算法实现。
我们提出一种受伯克维奇几何启发的非阿基米德空间优化新框架。具体引入多圆盘空间——即非阿基米德域上闭球的乘积，该空间在保持非阿基米德域刚性层级结构的同时，具备经典几何中缺失的优良性质。我们证明度量树可自然嵌入此类空间，表明其能有效表示分层数据。研究其度量几何，确立测地线唯一性，确认与经典优化方法兼容。进一步提出由多项式绝对值线性组合构成的一类实值函数，其在测地线上具分段多项式描述，并满足通用逼近性质。建立多圆盘空间上的优化理论：证明极小值存在性，并探索求解算法。配套提供开源 Julia 库，实现核心对象与优化流程。
原文摘要 · Abstract (English)
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.
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