Depth-induced bipolar neural collapse for privacy-preserving face verification

Neural Collapse (NC) is a foundational geometric phenomenon typically observed in the terminal phase of supervised deep network training, where class features converge toward highly symmetric, low-rank structures such as the simplex equiangular tight frame (ETF). While existing literature predominantly characterizes NC as a consequence of label-driven, gradient-based optimization, this work demonstrates that a structurally distinct regime, Bipolar Neural Collapse (BNC), can emerge in a label-free and training-free setting. We propose a fixed-weight multilayer transformation with the computational form of a feedforward network, using top-k selective routing and unit-norm projection on pretrained facial embeddings without learned weights or backpropagation. Our theoretical and empirical analyses reveal that this transformation introduces an inherent low-rank architectural bias that suppresses residual variance and progressively polarizes embeddings into anchor-specific antipodal clusters. The resulting stable and repeatable representations facilitate one-way cryptographic hashing for privacy-preserving face verification without requiring raw facial image storage, backbone retraining, or identity-specific code assignment. Evaluations on the LFW, CFP, and CMU-PIE benchmarks confirm that while moderate transformation depth improves genuine-pair consistency and verification utility, excessive depth induces over-collapse, suppresses inter-identity diversity, and increases pre-hash collision risk. These findings suggest that depth-induced BNC geometry provides a controllable training-free mechanism for balancing verification utility and privacy-preserving protected-template security.