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The Achilles' Heel of Angular Margins: A Chebyshev Polynomial Fix for Speaker Verification

Yang Wang, Yiqi Liu, Chenghao Xiao, Chenghua Lin

TL;DR

The paper addresses instability in angular-margin losses for speaker verification caused by the unbounded derivative of $\arccos$ as embeddings align with class prototypes. It introduces ChebyAAM, a stable alternative that replaces $\arccos$ with a Chebyshev polynomial approximation $f_{\mathrm{cheb}}(x,m)$, computed efficiently via Clenshaw's algorithm. Across VoxCeleb, SITW, and CN-Celeb, ChebyAAM consistently improves verification metrics (EER and mDCF) across multiple backbones, with degree 30 often yielding the best results. The work demonstrates that polynomial-angle approximations provide a robust, model-agnostic path for designing future metric-learning losses in speaker verification, reducing training instability while sharpening hard-example gradients.

Abstract

Angular margin losses, such as AAM-Softmax, have become the de facto in speaker and face verification. Their success hinges on directly manipulating the angle between features and class prototypes. However, this manipulation relies on the arccos function to recover the angle, introducing a significant yet overlooked source of training instability. The derivative of arccos explodes at its boundaries, causing gradient peaks during optimisation. Furthermore, the formulation fails to generate a sufficiently sharp gradient for hard-to-classify examples. We address these issues by proposing ChebyAAM, a loss that replaces the arccos operation with its Chebyshev polynomial approximation. This substitution eliminates gradient explosion and applies a stronger corrective signal to hard examples, leading to more effective optimisation. Experiments on three benchmarks (VoxCeleb, SITW, and CN-Celeb) demonstrate that our method resolves the instability and consistently improves performance. Our work suggests that approximating angular operations, rather than calculating them explicitly, offers a more robust path for designing future metric learning losses. Code is available at https://github.com/ExtraOrdinaryLab/vibe.

The Achilles' Heel of Angular Margins: A Chebyshev Polynomial Fix for Speaker Verification

TL;DR

The paper addresses instability in angular-margin losses for speaker verification caused by the unbounded derivative of as embeddings align with class prototypes. It introduces ChebyAAM, a stable alternative that replaces with a Chebyshev polynomial approximation , computed efficiently via Clenshaw's algorithm. Across VoxCeleb, SITW, and CN-Celeb, ChebyAAM consistently improves verification metrics (EER and mDCF) across multiple backbones, with degree 30 often yielding the best results. The work demonstrates that polynomial-angle approximations provide a robust, model-agnostic path for designing future metric-learning losses in speaker verification, reducing training instability while sharpening hard-example gradients.

Abstract

Angular margin losses, such as AAM-Softmax, have become the de facto in speaker and face verification. Their success hinges on directly manipulating the angle between features and class prototypes. However, this manipulation relies on the arccos function to recover the angle, introducing a significant yet overlooked source of training instability. The derivative of arccos explodes at its boundaries, causing gradient peaks during optimisation. Furthermore, the formulation fails to generate a sufficiently sharp gradient for hard-to-classify examples. We address these issues by proposing ChebyAAM, a loss that replaces the arccos operation with its Chebyshev polynomial approximation. This substitution eliminates gradient explosion and applies a stronger corrective signal to hard examples, leading to more effective optimisation. Experiments on three benchmarks (VoxCeleb, SITW, and CN-Celeb) demonstrate that our method resolves the instability and consistently improves performance. Our work suggests that approximating angular operations, rather than calculating them explicitly, offers a more robust path for designing future metric learning losses. Code is available at https://github.com/ExtraOrdinaryLab/vibe.
Paper Structure (13 sections, 7 equations, 2 figures, 3 tables)

This paper contains 13 sections, 7 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Motivation: The Instability of Arccos
  5. Proposed Polynomial Approximation
  6. Gradient of Chebyshev-based AAM-Softmax
  7. Experimental Setup
  8. Results and Discussion
  9. Main Results on VoxCeleb, SITW, and CN-Celeb
  10. Generalisability Across Architectures
  11. Ablation on Polynomial Degree
  12. Conclusion
  13. Acknowledgment and Disclosure

Figures (2)

  • Figure 1: Function, gradient (first derivative), and Hessian (second derivative) plots for AAM-Softmax ($\psi$) and its Chebyshev approximations ($f_{\mathrm{cheb}}$) with margin $m=0.3$. The polynomial versions offer smoother derivatives and bounded Hessians, avoiding the pathological spikes seen in the exact function near $x=\pm 1$.
  • Figure 2: Comparison of the derivative surfaces of N-Softmax (left), AAM-Softmax (middle), and Chebyshev-based AAM-Softmax (right) with respect to the true-class logit. The "hard" example (point A, where $s_p \approx s_n$) and the "easy" example (point B, where $s_p \ll s_n$) end up with similar magnitudes of gradients for the N-Softmax and AAM-Softmax losses, but differ more for the Chebyshev approach. Chebyshev AAM-Softmax can show a bigger derivative gap between A and B. It effectively can have a steeper slope near certain values, thus providing a higher penalty for "hard" examples relative to "easier" ones.