Schur-Complement Barrier Message Passing

Integrate this into a graph-neural-network or token-mixing block immediately before aggregation. Let $X\in\mathbb R^{N\times d}$ be node or token features, let $K\in\mathbb R^{N\times N}$ be a symmetric baseline affinity matrix produced from local attention, k-nearest-neighbor edges, or a fixed graph Laplacian, and let $E\in\mathbb R^{N\times m}$ assign each node to one of $m$ latent regions. Do not create persistent region tokens. Instead compute the Schur-complement correction per feature channel or, more cheaply, apply it to the scalar aggregation weights. Pseudocode: K = symmetric_affinity(X); E = soft_region_assignments(X); construct sparse barrier incidence rows in $B$ for selected region pairs; set D = softplus(raw_barrier_strength); compute A = E.T @ K @ E + B.T @ D @ B + eps*I; solve T = solve(A, E.T @ K @ X); form X_barrier = K @ X - K @ E @ T; then pass X_barrier through the usual value projection, residual connection, and nonlinearity. The implemented equation is $K_{\rm eff}X=KX-KEA^{-1}E^{\mathsf T}KX$. For a transformer, restrict $K$ to a local attention neighborhood and parameterize $E$ with a small router; for a GNN, use fixed communities initially and learn only barrier strengths. Estimate $K$ from current attention logits, while $E$, $B$, and $D$ are learned or sparsified; the solve is over $m\ll N$, so the added cost is $O(Nm^{2}+m^{3})$. Start with a 6-layer, 8-head graph transformer on Cora or synthetic stochastic-block graphs, comparing standard message passing, attention with a learned edge mask, and this module at equal hidden size. Measure validation accuracy, feature Dirichlet energy within and across planted communities, and oversmoothing via pairwise cosine similarity after six layers. Success is higher boundary-node accuracy and lower cross-community similarity without more than 10% wall-clock overhead; on long-depth stacks, test whether the Schur correction delays accuracy collapse relative to the baseline.