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

Graph Convolutional Networks often suffer from oversmoothing, where repeated message passing causes node representations to collapse into a uniform state. Sheaf Neural Networks address this by equipping graphs with stalk spaces and restriction maps, replacing the standard graph Laplacian with a sheaf Laplacian. Existing literature attempts to measure anti-oversmoothing capacity by looking at the absolute dimension of the harmonic space (the kernel of the sheaf Laplacian). However, the authors observe that a large absolute dimension can simply be the result of trivial channel-wise constant sections that still collapse node distinctions. This paper asks how to rigorously and relatively compare two sheaf frameworks to determine whether one genuinely possesses superior anti-oversmoothing capacity beyond trivial index inflation.

To solve the limitations of raw index counting, the authors develop a relative, geometric approach. They construct global cochain pairings, adjoint operators, and a discrete Hodge-theoretic framework that extends beyond Euclidean vector spaces. Because the raw index jump is Euler-balanced and insufficient on its own, they establish an index-theoretic comparison criterion (Theorem 5). This criterion combines the index jump with a degree-one heat trace correction and non-trivial holonomy conditions on the stalk transportation. When satisfied, it guarantees a strict geometric inclusion of harmonic spaces, meaning the target sheaf's harmonic space genuinely contains the reference sheaf's harmonic space beyond trivial inflation.

The paper extends this framework beyond linear vector spaces to non-linear geometries. When stalks are non-linear manifolds (such as Symmetric Positive Definite matrices), global algebraic tools break down because the coboundary operator ceases to be linear. The authors address this by introducing local tangent-space linearization and constructing a concrete non-linear instantiation called GyroSheaf. GyroSheaf equips stalks with a curved gyrovector-space structure and demonstrates that it admits a tangent Hodge decomposition, proving that the theoretical index criterion can successfully govern non-linear sheaf architectures.