Skeletal Prototypes on Iterative Nerve Expansions

Prototype-based neural networks aim to provide intrinsic interpretability by grounding predictions in a small set of part prototypes. However, modern vision backbones typically operate in normalized, directional embedding spaces where each semantic part exhibits substantial intra-class variability. As a result, point prototypes often become redundant or unstable, hurting both explanation quality and robustness. We propose vMFProto, a distributional part-prototype framework that models each class as a mixture of von Mises-Fisher components on the hypersphere. Each prototype learns its own concentration, capturing part-specific variability, and we use entropic optimal transport (OT) to obtain structured patch-to-prototype assignments. A two-stage training schedule performs OT-driven prototype discovery followed by end-to-end refinement with patch-level distillation and distribution-aware diversity regularization. Experiments with frozen DINO backbones show that vMFProto achieves leading consistency and distinctiveness on CUB-200-2011 and competitive classification accuracy across CUB, Stanford Dogs, and Stanford Cars. Qualitative results confirm that vMFProto yields localized, non-redundant part evidence.

Few-Shot Medical Image Segmentation (FSMIS) aims to delineate novel anatomical targets from one or a few annotated support images, addressing the annotation scarcity in medical imaging. Notwithstanding recent advancements, current prototype-based methods are bottlenecked by two coupled limitations: 1) cue entanglement, where a single spatial-domain prototype is forced to summarise organ silhouette, parenchymal texture and boundary appearance simultaneously, so any support-query mismatch on one cue propagates indiscriminately to the others; and 2) topology-blind matching, where cosine similarity measures distance in the ambient Euclidean space and ignores the connectivity of the underlying feature manifold, causing fragmented activations inside low-contrast organs and leakage into neighbouring tissues. To this end, we propose Spectral-Geodesic Prototype Network (SGP-Net), built around a Spectral-Geodesic Prototype Module with two coupled components. A Spectral Prototype Bank (SPB) decomposes support and query features into low-, mid- and high-frequency bands via learnable radial Fourier filters, yielding three disentangled prototypes per class that separately encode shape, texture and boundary cues. A Geodesic Matcher (GM) then replaces cosine similarity with a differentiable heat-diffusion approximation of geodesic distance, propagating matching signals along a feature affinity graph so that on-manifold pixels accumulate consistent responses while off-manifold look-alikes are suppressed. Experiments on three public FSMIS benchmarks demonstrate that SGP-Net achieves competitive performance against recent state-of-the-art methods.

Supervised classification has a theoretical optimum, Neural Collapse (NC), yet neither of its two dominant paradigms reaches it in practice. Cross entropy (CE) leaves radial degrees of freedom unconstrained and converges to a degenerate geometry, while supervised contrastive learning (SCL) drives features toward NC during pretraining but discards this structure in a post hoc linear probing phase. We show that both paradigms are different appearances of the same method that contrasts prototypes on the unit hypersphere, and that closing the gap requires fixing each at its point of failure. From the CE side, we propose NTCE and NONL, two normalized losses that import contrastive optimization's missing ingredients into classifier learning: a large effective negative set and decoupled alignment and uniformity terms. From the SCL side, we prove that SCL's objective already optimizes throughout training for a principled classifier whose weights are the class mean embeddings, making linear probing both redundant and harmful. Empirically, on four benchmarks including ImageNet-1K, NTCE and NONL surpass CE accuracy, closely approximate NC ($\geq 95\%$), and match CE's converged NC on 4/5 metrics in under $7.5\%$ of its iterations, while SCL with fixed prototypes matches linear probing without the hours-long classifier training phase. The learned geometry yields $+5.5\%$ mean relative improvement in transfer learning, up to $+8.7\%$ under severe class imbalance, and improved robustness to corruptions on ImageNet-C. Our work recasts supervised learning as prototype learning on the hypersphere, with NC reached by design.