Understanding Hyperspherical Geometry of ECAPA-TDNN Embedding and Its Impact on Zero-Shot Voice Conversion

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Angular-margin speaker encoders are widely used in voice conversion, yet the geometry of their classifier prototypes remains poorly understood. We analyze ECAPA-TDNN classifier prototypes as points on the unit hypersphere and characterize their organization using rotation-invariant angular statistics together with global and local effective dimensionality measures. Our analysis shows that standard training can induce angular concentration and a substantial reduction in effective dimensionality. To address this, we investigate two geometric regularization strategies (hinged Riesz log-energy and effective-dimension maximization) applied to classifier prototypes to encourage more uniform hyperspherical coverage. The resulting prototype sets exhibit higher effective dimensionality and improved isotropy, with configuration-dependent effects on speaker-recognition performance. When the corresponding ECAPA-TDNN models are used as speaker encoders for Fast-VGAN, the regularized systems also exhibit improved robustness in zero-shot voice conversion, particularly for previously unseen speakers.