提出在双曲球中进行聚类的新方法，为双曲机器学习打下基础。
Clustering in hyperbolic balls
- 基于新定义的质心提出双曲球上的k-means聚类
- 构建了用于双曲球概率混合模型的EM算法
- 适合对树状结构数据建模的研究者使用
将数据表示在负曲率流形中的思想近期受到广泛关注，催生了新的研究方向——双曲机器学习（ML）。为了充分发挥这一新范式的潜力，亟需高效的数据分析与统计建模技术。本文建立了双曲空间聚类的严格数学框架：首先，基于新的质心定义，提出了双曲球上的k-means聚类；其次，提出了用于学习双曲球中新型概率分布混合模型的期望最大化（EM）算法。由此奠定了双曲空间无监督学习的基础。
原文摘要 · Abstract (English)
The idea of representations of the data in negatively curved manifolds recently attracted a lot of attention and gave a rise to the new research direction named {\it hyperbolic machine learning} (ML). In order to unveil the full potential of this new paradigm, efficient techniques for data analysis and statistical modeling in hyperbolic spaces are necessary. In the present paper rigorous mathematical framework for clustering in hyperbolic spaces is established. First, we introduce the $k$-means clustering in hyperbolic balls, based on the novel definition of barycenter. Second, we present the expectation-maximization (EM) algorithm for learning mixtures of novel probability distributions in hyperbolic balls. In such a way we lay the foundation of unsupervised learning in hyperbolic spaces.
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