用群论框架提升超球空间机器学习的数学基础
A group-theoretic framework for machine learning in hyperbolic spaces
- 引入超球空间的均值（重心）与新型概率分布
- 提出高效算法计算重心与最大似然估计
- 为超球深度学习提供可扩展的理论工具
将数据嵌入超球空间能以极少维度保留复杂关系，从而实现紧凑模型并提升机器学习效率。其核心思想是：超球表示可避免特定类型数据的重要结构信息丢失。然而，超球机器学习的进一步发展需要更严谨的数学方法与充分的几何技术。本文旨在通过结合群论、共形几何、优化与统计方法，强化超球机器学习的数学基础。具体地，我们提出了超球体上的均值（重心）概念及一类新型概率分布，并设计了高效的重心计算与最大似然估计优化算法。这些基础概念可作为构建更复杂算法与实现超球深度学习流程的起点。
原文摘要 · Abstract (English)
Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic representations can prevent the loss of important structural information for certain ubiquitous types of data. However, further advances in hyperbolic ML require more principled mathematical approaches and adequate geometric methods. The present study aims at enhancing mathematical foundations of hyperbolic ML by combining group-theoretic and conformal-geometric arguments with optimization and statistical techniques. Precisely, we introduce the notion of the mean (barycenter) and the novel family of probability distributions on hyperbolic balls. We further propose efficient optimization algorithms for computation of the barycenter and for maximum likelihood estimation. One can build upon basic concepts presented here in order to design more demanding algorithms and implement hyperbolic deep learning pipelines.
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