This Looks Distinctly Like That: Grounding Interpretable Recognition in Stiefel Geometry against Neural Collapse

Prototype networks provide an intrinsic case based explanation mechanism, but their interpretability is often undermined by prototype collapse, where multiple prototypes degenerate to highly redundant evidence. We attribute this failure mode to the terminal dynamics of Neural Collapse, where cross entropy optimization suppresses intra class variance and drives class conditional features toward a low dimensional limit. To mitigate this, we propose Adaptive Manifold Prototypes (AMP), a framework that leverages Riemannian optimization on the Stiefel manifold to represent class prototypes as orthonormal bases and make rank one prototype collapse infeasible by construction. AMP further learns class specific effective rank via a proximal gradient update on a nonnegative capacity vector, and introduces spatial regularizers that reduce rotational ambiguity and encourage localized, non overlapping part evidence. Extensive experiments on fine-grained benchmarks demonstrate that AMP achieves state-of-the-art classification accuracy while significantly improving causal faithfulness over prior interpretable models.

Brain-HGCN: A Hyperbolic Graph Convolutional Network for Brain Functional Network Analysis

Functional magnetic resonance imaging (fMRI) provides a powerful non-invasive window into the brain's functional organization by generating complex functional networks, typically modeled as graphs. These brain networks exhibit a hierarchical topology that is crucial for cognitive processing. However, due to inherent spatial constraints, standard Euclidean GNNs struggle to represent these hierarchical structures without high distortion, limiting their clinical performance. To address this limitation, we propose Brain-HGCN, a geometric deep learning framework based on hyperbolic geometry, which leverages the intrinsic property of negatively curved space to model the brain's network hierarchy with high fidelity. Grounded in the Lorentz model, our model employs a novel hyperbolic graph attention layer with a signed aggregation mechanism to distinctly process excitatory and inhibitory connections, ultimately learning robust graph-level representations via a geometrically sound Fr\'echet mean for graph readout. Experiments on two large-scale fMRI datasets for psychiatric disorder classification demonstrate that our approach significantly outperforms a wide range of state-of-the-art Euclidean baselines. This work pioneers a new geometric deep learning paradigm for fMRI analysis, highlighting the immense potential of hyperbolic GNNs in the field of computational psychiatry.