Speaker embedding learning based on Euclidean space has achieved significant progress, but it is still insufficient in modeling hierarchical information within speaker features. Hyperbolic space, with its negative curvature geometric properties, can efficiently represent hierarchical information within a finite volume, making it more suitable for the feature distribution of speaker embeddings. In this paper, we propose Hyperbolic Softmax (H-Softmax) and Hyperbolic Additive Margin Softmax (HAM-Softmax) based on hyperbolic space. H-Softmax incorporates hierarchical information into speaker embeddings by projecting embeddings and speaker centers into hyperbolic space and computing hyperbolic distances. HAM-Softmax further enhances inter-class separability by introducing margin constraint on this basis. Experimental results show that H-Softmax and HAM-Softmax achieve average relative EER reductions of 27.84% and 14.23% compared with standard Softmax and AM-Softmax, respectively, demonstrating that the proposed methods effectively improve speaker verification performance and at the same time preserve the capability of hierarchical structure modeling. The code will be released at https://github.com/PunkMale/HAM-Softmax.

翻译：基于欧几里得空间的说话人嵌入学习已取得显著进展，但在建模说话人特征内部的层级信息方面仍显不足。双曲空间凭借其负曲率几何特性，能够在有限体积内高效表示层级信息，从而更契合说话人嵌入的特征分布。本文提出基于双曲空间的Hyperbolic Softmax（H-Softmax）与Hyperbolic Additive Margin Softmax（HAM-Softmax）。H-Softmax通过将嵌入向量与说话人中心投影至双曲空间并计算双曲距离，将层级信息融入说话人嵌入中。HAM-Softmax在此基础上引入间隔约束，进一步增强类间可分性。实验结果表明，与标准Softmax和AM-Softmax相比，H-Softmax和HAM-Softmax分别实现了平均相对EER降低27.84%和14.23%，证明所提方法在有效提升说话人验证性能的同时，保留了层级结构建模能力。代码发布于https://github.com/PunkMale/HAM-Softmax。