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

TL;DR

This paper introduces ChebyAAM, a novel loss function for speaker verification that replaces the arccos operation with a Chebyshev polynomial approximation, improving training stability and performance.

Contribution

It proposes a Chebyshev polynomial-based approximation to replace arccos in angular margin losses, addressing gradient explosion and enhancing hard example learning.

Findings

Resolves gradient explosion issues in angular margin losses.

Achieves consistent performance improvements on VoxCeleb, SITW, and CN-Celeb.

Provides a robust alternative for future metric learning loss design.

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.