Unlocking Better Closed-Set Alignment Based on Neural Collapse for Open-Set Recognition

Chaohua Li, Enhao Zhang, Chuanxing Geng, Songcan Chen

Abstract: In recent Open-set Recognition (OSR) community, a prevailing belief is that enhancing the discriminative boundaries of closed-set classes can improve the robustness of Deep Neural Networks (DNNs) against open data during testing. Typical studies validate this *implicitly* by empirical evidence, without a formalized understanding of *how DNNs help the closed-set features obtain more discriminative boundaries?* For this, we provide an answer from the Neural Collapse (NC) perspective: DNNs align the closed-set with a *Simplex Equiangular Tight Frame* (ETF) structure that has geometric and mathematical interpretability. Regrettably, although NC naturally occurs in DNNs, we discover that typical studies cannot guarantee the features being learned to strictly align with the ETF. Thus, we introduce a novel concept, Fixed ETF Template (FiT), which holds an ideal structure associated with closed-set classes. To force class means and classifier vectors to align with FiT, we further design a Dual ETF (DEF) loss involving two components. Specifically, *F*-DEF loss is designed to align class means with FiT strictly, yielding optimal inter-class separability. Meanwhile, we extend a dual form to classifier vectors, termed *C*-DEF loss, which guides class means and classifier vectors to satisfy self-duality. Our theoretical analysis proves the validity of the proposed approach, and extensive experiments demonstrate that DEF achieves comparable or superior results with reduced computational resources on standard OSR benchmarks.