hierarchical sheaf pooling

Existing sheaf neural networks lack an effective hierarchical pooling mechanism for multiscale graph modeling. This work proposes HiSP, a framework that introduces, for the first time, a learnable sheaf-aware hierarchical pooling scheme. Built upon local spectral coarsening, HiSP projects fine-grained stalk features onto low-frequency modes of the cluster-wise sheaf Laplacian and preserves sheaf energy consistency via Galerkin operators. The method integrates cochain-level prolongation maps with a lifting-based sheaf Laplacian, enabling efficient batch processing within PyTorch Geometric. HiSP not only explicitly distinguishes and quantifies truncation and realization losses incurred during coarsening but also significantly enhances multiscale representation capacity while preserving the underlying sheaf structure.

This study addresses the over-smoothing problem in deep neural networks and the unreliability of existing harmonic space dimension metrics by proposing a relative geometric criterion grounded in index theory to precisely characterize over-smoothing resistance. Pioneering a relative geometric analysis perspective, this work integrates local tangent space linearization with the stalk structure of curved gyrovector spaces to construct a nonlinear gyrosheaf diffusion model, thereby overcoming the limitations of absolute dimensionality. Extensive experiments across ten architectures validate the effectiveness of the proposed criterion. Notably, the gyrosheaf diffusion model successfully maintains representation stability and prevents collapse in deep networks, confirming the theoretical validity of this approach. These findings establish a robust framework for analyzing and mitigating over-smoothing through rigorous geometric principles rather than conventional spectral methods.

Existing neural layer diffusion models rely on SVD-based normalization and dense edge-restriction mappings, leading to high computational overhead, gradient instability, and performance degradation as stalk dimension increases. To address these issues, we propose Polynomial Neural Stratified Diffusion (PolyNSD): a spectral diffusion framework leveraging K-th order orthogonal polynomials of the normalized layer Laplacian to enable explicit K-hop propagation over cell complexes. PolyNSD introduces trainable convex-combination spectral responses, spectral rescaling, and residual/gated pathways, while employing only diagonal restriction mappings—thereby decoupling model performance from stalk dimension. This design effectively mitigates heterophily and oversmoothing, enhancing training stability and inference efficiency. Evaluated on both homophilic and heterophilic graph benchmarks, PolyNSD achieves state-of-the-art accuracy with significantly reduced memory consumption and runtime, eliminating reliance on SVD decomposition and dense parameterization.

This work proposes a simplex-based multidimensional persistence Laplacian framework to address the sensitivity of dimensionality reduction methods like Principal Component Analysis (PCA) to the choice of target dimension and their resulting performance instability. The approach introduces, for the first time, the multidimensional persistence Laplacian into image analysis by integrating multiscale topological spectra across multiple reduced dimensions and employing statistical aggregation to produce robust feature representations. Experimental results on the COIL20 and ETH80 datasets demonstrate that the proposed method significantly outperforms PCA baselines at moderate dimensions and exhibits greater stability across varying target dimensions, thereby enhancing both the discriminability and robustness of image representations.

To address the limited expressive power of conventional hypergraphs in modeling higher-order relationships, this paper introduces the sheaf structure—previously unexplored in hypergraph learning—yielding the sheaf hypergraph model. We define both linear and nonlinear sheaf hypergraph Laplacians to explicitly encode local higher-order dependencies between nodes and hyperedges, thereby endowing the model with stronger inductive biases. Building upon this, we propose the Sheaf Hypergraph Neural Network (SheafHGNN), which integrates sheaf-based convolution with message-passing mechanisms. Extensive experiments on multiple hypergraph node classification benchmarks demonstrate that SheafHGNN consistently outperforms state-of-the-art methods, validating the effectiveness and generalization advantage of sheaf structures for capturing higher-order topological patterns.

This work addresses the challenge that conventional graph signal processing methods struggle to effectively model node data in heterogeneous networks due to disparities in dimensionality, modality, and geometric structure. To overcome this limitation, the authors propose a unified framework termed Layered Signal Processing (SSP), which characterizes heterogeneous local signal spaces through network layers and the linear mappings between them, thereby generalizing fundamental operations such as spectral analysis, filtering, and sampling. Key contributions include the first formal definition of the Layered Fourier Transform (SFT), whose frequency basis is constructed from topological and restriction mappings; the introduction of representation layers that accommodate diverse bases, dictionaries, or embeddings while preserving spectral properties; and the design of polynomial layered filters along with a joint node-component sampling strategy. Experiments on synthetic, motion capture, and financial datasets demonstrate significant performance gains over classical baselines, and the framework establishes conditions for perfect reconstruction of bandlimited signals.

This work addresses the limitations of traditional unsupervised community detection methods on heterophilic graphs, where connected nodes often belong to different classes, and existing approaches either ignore node features or rely on opaque mechanisms. To overcome these challenges, the authors propose Curvature-Guided Stratified Diffusion (CGSD), a novel framework that uniquely leverages discrete Forman–Ricci curvature as the sole topological signal throughout an end-to-end unsupervised pipeline. Key innovations include a curvature-gated stratified diffusion encoder, a curvature-aware spectral clustering module (CSpec), and the first unified benchmark tailored for fully unsupervised methods. Evaluated on five heterophilic graph benchmarks, CGSD substantially outperforms nine baselines, achieving particularly strong results on Wisconsin and Chameleon datasets. CSpec yields a 15% average improvement in Normalized Mutual Information (NMI, p=0.008), and the curvature distribution demonstrates high interpretability.

This work addresses the challenge of efficiently and accurately clustering temporal graphs to obtain coarse-grained representations when node attributes are missing or weak. The authors propose a theory-driven temporal graph pooling method that formulates community detection as a pooling operator grounded in spectral graph theory. By integrating graph neural networks with multi-slice modularity optimization and incorporating GPU-accelerated spectral clustering and stochastic block models, the approach yields a scalable clustering primitive that remains effective even in attribute-scarce settings. Empirical results demonstrate that while the method excels in scenarios lacking strong attribute signals, neural models achieve superior performance when structural, temporal, and attribute information align coherently. This study thus establishes a new pathway for temporal graph coarsening that balances theoretical rigor with computational efficiency.

Existing approaches struggle to capture the global organization of local interactions among nodes in vector fields defined over graphs. This work proposes SheafIQ, a novel framework that uniquely integrates sheaf theory and information theory to map vectors from adjacent nodes into a unified edge-associated coordinate system. By analyzing the residual energy distribution and its entropy, SheafIQ quantifies the overall orderliness of the vector field. The method transcends conventional limitations that focus solely on graph topology or nodal signals, offering a unified information-theoretic measure for vector-valued states on geometric graphs. Applied to diverse systems—including protein–protein interaction networks, functional brain connectomes, urban traffic flows, and power grids—SheafIQ uncovers complementary organizational information beyond what classical graph- and signal-based metrics reveal.

This study addresses the failure of long-range information propagation and the over-smoothing bottleneck in graph neural networks by proposing the ONDA framework. To the best of our knowledge, this work is the first to couple wave dynamics with matrix-valued transport, leveraging operator-valued information waves to facilitate long-range graph learning. By introducing second-order oscillatory dynamics alongside layered transport operators, ONDA ensures that cross-node influence remains undiminished during propagation, thereby overcoming the fundamental limitations of conventional diffusion models. Extensive experiments demonstrate that ONDA significantly outperforms existing state-of-the-art methods across diverse benchmark tasks, including long-range propagation, graph transfer learning, and heterogeneous graph modeling.