arXiv subjects

Junwen Dong

Publications and source records attributed to Junwen Dong.

3 recordsLinked to original sources

arXiv2026

Geometric Structures on Graphs: a Holonomy-Based Discretization of Curvature

We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric $g_i$, and each directed edge carries a reversible metric-compatible transport $F_{ij}$. The ordered product around an oriented triangular loop $\mathcal C$ gives a holonomy $H_{\mathcal C}$, whose normalized logarithm $\Omega_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})$ is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since $\Omega_{\mathcal C}$ lies in the $g_i$-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy--curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.

cs.LG↗

arXiv2026

Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion

To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce \textit{GyroSheaf}, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.

cs.LG↗

arXiv2026

Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

cs.LG↗