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.

翻译：角间隔损失函数（如AAM-Softmax）已成为说话人与人脸验证领域的事实标准。其成功关键在于直接操纵特征向量与类别原型之间的夹角。然而，这种操作依赖反余弦函数来恢复角度，这引入了一个重要但被忽视的训练不稳定性来源。反余弦函数在其边界处的导数会发散，导致优化过程中出现梯度峰值。此外，该公式难以对难以分类的样本产生足够尖锐的梯度信号。我们通过提出ChebyAAM损失函数来解决这些问题，该函数使用切比雪夫多项式逼近替代原有的反余弦运算。这种替换消除了梯度爆炸现象，并对困难样本施加更强的校正信号，从而实现更有效的优化。在三个基准数据集（VoxCeleb、SITW和CN-Celeb）上的实验表明，我们的方法解决了不稳定性问题并持续提升性能。本研究指出，对角度运算进行逼近而非显式计算，为设计未来度量学习损失函数提供了更稳健的路径。代码发布于https://github.com/ExtraOrdinaryLab/vibe。