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Hyperbolic Additive Margin Softmax with Hierarchical Information for Speaker Verification

Zhihua Fang, Liang He

TL;DR

This paper tackles the limitation of Euclidean speaker embeddings in capturing hierarchical information by introducing hyperbolic losses for speaker verification. It proposes Hyperbolic Softmax (H-Softmax) and Hyperbolic Additive Margin Softmax (HAM-Softmax), which project embeddings and class centers into the Poincaré ball and use the hyperbolic distance to define logits; HAM-Softmax adds a margin to explicitly enlarge inter-class separation. Across VoxCeleb1/2 and CNCeleb, the methods achieve substantial gains, with average relative EER reductions of 27.84% for H-Softmax and 14.23% for HAM-Softmax compared to Softmax and AM-Softmax, respectively, and show that high-curvature hyperbolic space effectively models hierarchical structure. Ablation studies clarify the roles of curvature, scale, and margin, yielding practical guidelines for leveraging hyperbolic geometry in speaker embedding learning.

Abstract

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.

Hyperbolic Additive Margin Softmax with Hierarchical Information for Speaker Verification

TL;DR

This paper tackles the limitation of Euclidean speaker embeddings in capturing hierarchical information by introducing hyperbolic losses for speaker verification. It proposes Hyperbolic Softmax (H-Softmax) and Hyperbolic Additive Margin Softmax (HAM-Softmax), which project embeddings and class centers into the Poincaré ball and use the hyperbolic distance to define logits; HAM-Softmax adds a margin to explicitly enlarge inter-class separation. Across VoxCeleb1/2 and CNCeleb, the methods achieve substantial gains, with average relative EER reductions of 27.84% for H-Softmax and 14.23% for HAM-Softmax compared to Softmax and AM-Softmax, respectively, and show that high-curvature hyperbolic space effectively models hierarchical structure. Ablation studies clarify the roles of curvature, scale, and margin, yielding practical guidelines for leveraging hyperbolic geometry in speaker embedding learning.

Abstract

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.
Paper Structure (14 sections, 8 equations, 3 figures, 2 tables)

This paper contains 14 sections, 8 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Discriminative Loss for Speaker Recognition
  4. Hyperbolic Geometry in Deep Learning
  5. Methodology
  6. Preliminaries
  7. Hyperbolic Softmax
  8. Hyperbolic Additive Margin Softmax
  9. Experiments
  10. Datasets and Experiments Setup
  11. Results on VoxCeleb 1&2 and CNCeleb
  12. Ablation Study
  13. Conclusions
  14. Acknowledgements

Figures (3)

  • Figure 1: Illustration of the hierarchical structure of speaker features and hyperbolic space. Left: speaker features contain hierarchical information (e.g., fundamental frequency, formant structure, and prosody, etc.). Right: hyperbolic space with negative curvature geometry is more suitable for representing such hierarchical structures.
  • Figure 2: The framework of the proposed method, which incorporates the hyperbolic loss $\mathcal{L}_{\text{H-Softmax}}$ and the hyperbolic additive margin loss $\mathcal{L}_{\text{HAM-Softmax}}$.
  • Figure 3: EER (%) results of the curvature ablation study on VoxCeleb1 (without data augmentation).