Sheaf Compatibility Robustness Loss

Integrate the loss after a graph or simplicial neural network produces local features \(x_\sigma\). For a graph, use vertices as 0-simplices and edges as 1-simplices; for a triangle complex, include faces as 2-simplices. Give each incidence \(\sigma\subseteq\tau\) a fixed or learned linear restriction map \(F_{\sigma\to\tau}\), initialized as a projection from the lower-dimensional feature space to the higher-dimensional one. Construct \(K+1\) degraded subcomplexes by independently deleting a prescribed fraction of edges or triangles while retaining valid face incidences. For each training example, compute features on the full complex and each masked subcomplex. For every level \(k\), accumulate \(r_k=\sum_{\sigma\subseteq\tau\in X^k}w_{\sigma\tau}\|F_{\sigma\to\tau}x_\sigma-x_\tau\|_2^2\), then optimize task loss plus the average \(r_k\) and the prediction-consistency term in the displayed formula. Enforce functoriality by parameter tying for composable maps or by adding sampled penalties \(\|F_{\tau\to\mu}F_{\sigma\to\tau}-F_{\sigma\to\mu}\|_F^2\). Map composition and global-section equations come from the paper; deletion masks, weights, and \(\lambda,\mu\) are empirical hyperparameters. First test on Cora or a small triangle mesh with a 3-layer GNN, comparing ordinary message passing against this objective under 10%, 25%, and 40% random edge or face deletion. Success means unchanged clean accuracy, improved degraded accuracy, lower prediction variance across masks, and reduced incidence residuals, with overhead scaling linearly in the number of incidences.