HIERARCHICAL SPECTRAL LEARNING: LOCALITY BEFORE FREQUENCY FOR GRAPH NEURAL NET- WORKS

Graph neural networks on heterophilous graphs face two coupled challenges: neighborhood aggregation can mix semantically incompatible nodes, and a single latent geometry may fail to capture the coexistence of hierarchical, flat, and cyclic graph structure. Existing methods typically address only one of them: geometric GNNs improve representation fidelity in curved spaces but still rely on smoothing-based aggregation, while heterophily-aware spectral methods improve signal separation yet ignore the geometric mismatch between node representations that live in different tangent spaces. We propose a Curvature-Transport Spectral Network (CTS-Net) that resolves these limitations by combining mixed-curvature representation learning with transport-aware and track-specific propagation. CTS-Net embeds nodes in a mixed-curvature product manifold to model heterogeneous graph geometry, parallel-transports neighbor messages to the receiver tangent space to align cross-node representations before aggregation, routes the aligned messages into smooth and contrastive tracks through a learned orthogonal operator to separate compatible from incompatible interactions, and applies track-specific polynomial diffusion to learn distinct spectral responses for the two types of signals. Our analysis shows that orthogonal routing preserves inter-track separation and that transport avoids additional distortion during aggregation. Across ten heterophilous node-classification benchmarks, CTS-Net achieves the best average performance among the compared baselines while remaining efficient in memory and runtime.