A HYPERSPHERICAL–MARGIN FRAMEWORK FOR COMPOUND FACIAL EMOTIONS

Abstract

We propose a hyperspherical–margin framework for compound facial emotion recognition. Let  denote images and  compound classes. Standard softmax training optimizes likelihood but does not enforce angular separation on the representation manifold, yielding entangled decision regions and poorly calibrated scores. We therefore place FER on the unit hypersphere by L2-normalizing features and class weights, and train with an angular-margin head (CosFace/ArcFace) using a stability-minded three-stage curriculum: warm-up (), head-only margin with a gentle ramp, and partial fine-tuning of the upper backbone. To link geometry to operation, we evaluate (i) symmetric confusion mass for class pairs, (ii) macro one-vs-rest ROC/PR, and (iii) calibration via reliability diagrams and Expected Calibration Error (ECE). On RAF-DB compound-11, the proposed regimen improves macro metrics from a softmax warm-up to the final model as follows: macro-F1 , ROC-AUC , and PR-AUC . Probability calibration markedly improves (ECE , ≈−89.9%). Pairwise confusions collapse for historically adjacent categories; for example, Fearfully Angry ↔ Fearfully Surprised reduces from 96 to 2 total errors. The model remains lightweight (≈3.25M parameters; ≈0.43 GFLOPs at ) and edge-capable (GTX 1650 Max-Q: ~7.3 ms median, ~137 FPS; CPU: ~11.1 ms, ~90 FPS; batch=1). These results support a geometry-first thesis for compound FER: enforcing hyperspherical angular margins, coupled with a gentle training schedule, yields separable, calibrated embeddings without increasing backbone capacity.